elma Rd ¢ a CTR Ps S Lo { A : EPIEIIERS Ce Rit ise ee Sons PR SRERGE hth) 5 bir IAAL TS ge TaiyReiee SEE PY - DREN XC nd He ry TAT EA Wop : a RNA F KR APAS RRA IRR 4 Ged i FOR SOUR BR i . ¢ ENSURE IS y p : io ! p ¥ SAE ee pheline esreeasete BERKELEY LIBRARY UNIVERSITY OF CALIF A ee STRONG py Usiggy THEORY AND APPLICATION OF INFINITE SERIES BY DR. KONRAD KNOPP PROFESSOR OF MATHEMATICS AT THE UNIVERSITY OF TUBINGEN Translated from the Second German Edition by Miss R. C. Young, L. és Se. HAFNER PUBLISHING COMPANY NEW YORK : ASTRONOMY LIBRARY Printed in Great Britain by Blackie & Son, Ltd., Glasgow ASTRONOMY LIBRARY Preface to the first (German) edition. | There is no general agreement as to where an account of the theory of infinite series should begin, what .its main outlines should be, or what it should include. On the one hand, the whole of higher analysis may be regarded as a field for the application of this theory, for all limiting processes — including differentiation and integration — are based on the investigation of infinite sequences or of infinite series. On the other hand, in the strictest (and therefore narrowest) sense, the only matters that are in place in a textbook on infinite series are their definition, the manipulation of thc symbolism connected with them, and the theory of convergence. In his “Vorlesungen iiber Zahlen- und Funktionenlehre”, Vol. 1, Part. 2, A. Pringsheim has treated the subject with these limitations. There was no question of offering anything similar in the present book. My aim was quite different: namely, to give a comprehensive account of all the investigations of higher analysis in which infinite series are the chief object of interest, the treatment to be as free from assumptions as possible and to start at the very beginning and lead on to the extensive frontiers of present-day research. To set all this forth in as interesting and intelligible a way as possible, but of course without in the least abandoning exactness, with the object of providing the student with a convenient introduction to the subject and of giving him an idea of its rich and fascinating variety — such was my vision. The material grew in my Tends, however, and resisted my efforts to put it into shape. In order to make a convenient and useful book, the field had to be restricted. But I was guided throughout by the experience I have gained in teaching — I have covered the whole of the ground several times in the general course of my work and in lectures at the universities of Berlin and Konigsberg — and also by the aim of the book. It was fo give a thorough and reliable treatment, which would be of assistance to the student attending lectures and which Would at the same time be adapted for private study. ~The latter aim was particularly dear to me, and this accounts d the form in which I have presented the subject-matter. Since it : : NCIS 748 2 VI Preface. is generally easier — especially for beginners — to prove a deduction in pure mathematics than to recognize the restrictions to which the train of reasoning is subject, I have always dwelt on theoretical diffi- culties, and have tried to remove them by means of repeated illustra- tions; and although I have thereby deprived myself of a good deal of space for important matter, I hope to win the gratitude of the student. I considered that an introduction to the theory of real numbers was indispensable as a beginning, in order that the first facts relating to convergence might have a firm foundation. To this introduction I have added a fairly extensive account of the theory of sequences, and, finally, the actual theory of infinite series. The latter is then con- structed in two storeys, so to speak: a ground-floor, in which the classical part of the theory (up to about the stage of Cauchy's Analyse algébrique) is expounded, though with the help of very limited resources, and a super-structure, in which I have attempted to give an account of the later developments of the 19%h century. For the reasons mentioned above, I have had to omit many parts of the subject to which I would gladly have given a place for their own sake. Semi-convergent series, Euler's summation formula, a de- tailed treatment of the Gamma-function, problems arising from the hypergeometric series, the theory of double series, the newer work on power series, and, in particular, a more thorough development of the last chapter, that on divergent series — all these I was reluctantly obliged to set aside. On the other hand, I considered that it was essential to deal with sequences and series of complex terms. As the theory runs almost parallel with that for real variables, however, I have, from the beginning, formulated all the definitions and proved all the theorems concerned in such a way that they remain valid without alteration, whether the “arbitrary” numbers involved are real or com- plex. These definitions and theorems are further distinguished by the sign ©. In choosing the examples — in this respect, however, I lay no claim to originality; on the contrary, in collecting them I have made extensive use of the literature — I have taken pains to put practical applications in the fore-front and to leave mere playing with theoretical niceties alone. Hence there are e. g. a particularly large number of exercises on Chapter VIII and only very few on Chapter IX. Unfort unately there was no room for solutions or even for hints for the solution of the examples. A list of the most important papers, comprehensive accounts, and textbooks on infinite series is given at the end of the book, imme- diately in front of the index. Preface. : VII My friends Herren R. Courant, E. Jacobsthal, H. Rademacher, and H. Vermeil have assisted me in reading the proofs. I am specially grateful to Herren Jacobsthal and Rademacher for much advice and numerous improvements both in the subject-matter and in its arrange- ment. Here I wish to express my very hearty thanks to them all once more. I should also like to take this opportunity of thanking the pub- lishers, who have given their services to Mathematics in such a broad minded way in these difficult times, and who have also entered willingly into all my wishes during the printing of this book. Konigsberg, September 1921. Konrad Knopp. Preface to the second edition. The fact that a second edition was called for after such a remarkably short time could be taken to mean that the first had on the whole been on the right lines. Hence the general plan has not been altered, but it has been improved in the details of expression and demonstration on almost every page. Somewhat greater changes or additions are to be found in particular on pp. 34, 38, 45, 96 seqq., 126 seqq., 186 seqq., 193 seq., 204 seqq., 210 seq., 282, 322 seqq., 337 seq., 359 seq., 372, 429—433. The last chapter, that dealing with divergent series, has been wholly rewritten, with important extensions, so that it now in some measure provides an introduction to the theory and gives an idea of modern work on the subject. Dy. F. Lettenmeyer has taken great pains in assisting me to carry out the revision and read the proofs of the new edition. Konigsberg, December 1923. Konrad Knopp. Preface to the English edition. This translation of the second German edition has been very skilfully prepared by Miss R. C. Young, L. ¢s Sc. (Lausanne), Research Student, Girton College, Cambridge. The publishers, Messrs. Blackie and Son, Ltd., Glasgow, have carefully superintended the printing. In addition, the publishers were kind enough to ask me to add a chapter on Euler's summation formula and asymptotic expansions. I agreed to do so all the more gladly because, as I mentioned in VIII Preface. the original preface, it was only with great reluctance that I omitted this part of the subject in the German edition. This chapter has been translated by Miss W. M. Deans, B. Sc. (Aberdeen), B. A. (Cantab.), with equal skill. » I wish to take this opportunity of thanking the translators and the publishers for the trouble and care they have taken. If — as I hope — my book meets with a favourable reception and is found useful by English-speaking students of Mathematics, the credit will largely be theirs. Tiibingen, February 1928. Konrad Knopp. Contents. Introduction Part 1. Real numbers and sequences. Chapter], Principles of the theory of real numbers. ..The system of rational numbers and its gaps. . . . . . . . Sequences. of rational nuMDErS .... i. hl. iw she ie ie, . Irrational numbers . Completeness and rintqueness of the evston of real Wndiere COR COR OR COG COR CU HS QO DO = . Radix fractions and the Dedelind section . 0% (20 Ji. oe, Exerciseson Chapter 1 (1—8) .'v LJ iL, Te a. Chapter II. Sequences of real numbers. 8 6. Arbitrary sequences and arbitrary null sequences. . . . ... ,. § 17. Powers, roots, and logarithms. Special null sequences . § 8. Convergent sequences . : 8 9: The two main criteria . '. . . . poe ew § 10. Limiting points and upper and Tower Hits § 11. Infinite series, infinite products, infinite and contipned fractiors Exercises on Chapter 1 (9—383) ...:. .. . . S eieh vok for Tie she he Part 11. Foundations of the theory of infinite series. Chapter NL Series of positive terms. § 12. The first principal criterion and the two comparison tests lie § 13. The root test and the ratio test . . , . Gla) co Hei sue ne ht § 14. Series of positive, monotone decreasing terms . vein eee Ws Exercises on Chapter TIL (8444) Lor. oo 0 ooo Witte ian Page. 12 22 32 41 49 62 76 87 98 106 110 116 120 125 . Products with positive terms . y . Products with arbitrary terms. Absolute converdante . Connection between series and prose Conditional and wconditional Contents. Chapter IV. Series of arbitrary terms. . The second principal criterion and the algebra of convergent series . Absolute convergence. Derangement of series . . Multiplication of infinite serfes =. . . . . . . . «so . s & Exercises on Chapter IV (45—63) . Chapter V, Power series. . The radius of convergence . . Functions of a real variable . ; . Principal properties of functions repratented By ower ¢ series . . The algebra of power series . Exercises on Chap er V (64-73) Chapter VL The expansions of the so-called elementary functions. . The rational functions . § 23. § 24. § 25. § 26. . The cyclometrical functions he exponential function 2.0 0 as ahh iid eae ee The trigonometrical functions The binomial series . The logarithmic series Byercises on Chapter VI(74--B4, .-. . , , ... . .. .. Chapter VIL Infinite products. elite ie im. convergence Exercises on Chapter vil (85— 99) Chapter VIIL Closed and numerical expressions for the sums of series. . Statement of the problem . Evaluation of the sum of a series by 1 means ‘of a closed expression . Transformation of series . . Numerical evaluations . . Applications of the transformation of series “to pumerical evaluations Exercises on Chapter VIII (100—132) . eT Page. 126 . 136 3 346: . 149 151 158 171 179 188 189 191 208 211 213 215 218 221 226 228 230 232 240 247 260 267 § 36. § 37. § 88. § 39. § 40. § 41. § 42. § 43. § 44. § 45. § 46. $47. § 48. § 49. < Contents. Part 101 Development of the theory. Chapter IX. Series of positive terms. Detailed study of the two comparison tests . The logarithmic scales ia Special tests of the second kind . Theorems of Abel, Dini, and Pringsheim, and their application % a fresh deduction of the logarithmic scale of comparison tests Series of monotonely diminishing terms . ; General remarks on the theory of the convergence andl divargsnce of series of positive terms . Systematization of the general fens of convergence : Exercises on Chapter IX (133—141) . Chapter X. Series of arbitrary terms. Tests of convergence for series of arbitrary terms Rearrangement of conditionally convergent series Multiplication of conditionally convergent series . Exercises on Chapter X (142—153) . Chapter XI. Series of variable terms (Sequences of functions). Uniform convergence ; Passage to the limit term by ei : Tests of uniform convergence Fourier series . . A. Euler's formulae B. Dirichlet's integral C. Conditions of convergence . Applications of the theory of Fourier series . § 51. Products with variable terms . . Exercises on Chapter XI (154— 173) . Chapter XII Series of complex terms. Complex numbers and sequences . . Series of complex terms . Power series. Analytic functions . XI Page. 274 278 284 290 298 298 305 311 312 318 320 324 326 338 344 350 350 364 372 380 885 388 396 401 XII 4 Contents. § BS. The elementary analytic functions . .. . , « ois + « +o 0 vo — . Rational functions II. The exponential function III. The functions cosz and sinz IV. The functions cotz and tanz V. The logarithmic series VI. The inverse sine series VII. The inverse tangent series VIII. The binomial series . § 56. Series of variable terms. Uniform convergence. Weierstrass’ theo- rem on double series ciel $ 57. Products with complex terms .. . . SE hy aT § 58. Special classes of series of analytic Suactions bi bie ieee sie A. Divichlets series. « Lai wn i on wee. B. Faculty series, 2s 0.0. di sides C. Lambert's series . Exercises on Chapter XII (174-190) Chapter XIII. Divergent series. . General remarks on divergent series and the processes of limitation . The C- and H- processes ‘ . . Application of C,- summation to the theory of Toutions series . . The A- process , . The E- process . Exercises on Chapter XIII (200— 216) . OR COR CR UD DOH DO Ut wo HO © Chapter XIV. Euler's summation formula and asymptotic expansions. § 64. Euler's summation formula . A. The summation formula. . B. Important applications C. The evaluation of remainders . . Asymptotic series . : . Special cases of asymptotic espaisiond 3 won oR DD oS On A. Examples of the expansion problem B. Examples of the summation problem . Exercises on Chapter XIV oi Zon hl Bibliography: . . ". . . . eee Name and subject Indes oe a 428 441 441 446 448 452 457 Introduction. The foundation on which the structure of higher analysis rests is the theory of real numbers. Any strict treatment of the foundations of the differential and integral calculus and of related subjects must in- evitably start from there; and the same is true even for e. g. the cal- culation of roots and logarithms. The theory of real numbers first creates the material on which Arithmetic and Analysis can subsequently build, and with which they deal almost exclusively. The necessity for this has not always been realized. The great creators of the infinitesimal calculus — Leibniz and Newfon® — and the no less famous men who developed it, of whom Euler? is the chief, were too intoxicated by the mighty stream of learning springing from the newly-discovered sources to feel obliged to criticize fundamentals. To them the results of the new methods were sufficient evidence for the security of their foundations. It was only when the stream began to ebb that critical analysis ventured to examine the fundamental con- ceptions. About the end of the 18th century such efforts became stronger and stronger, chiefly owing to the powerful influence of Gauss ®. Nearly a century had to pass, however, before the most essential matters could be considered thoroughly cleared up. Nowadays rigour in connection with the underlying number concept is the most important requirement in the treatment of any mathematical subject. Ever since the later decades of the past century the last word on the matter has been uttered, so to speak, — by Weierstrass* in the sixties, and by Cantor® and Dedekind® in 1872. No lecture or treatise 1 Gottfried Wilhelm Leibniz, born in Leipzig in 1648, died in Hanover in 1716. Isaac Newton, born at Woolsthorpe in 1642, died in London in 1727. Each discovered the foundations of the infinitesimal calculus independently of the other. 2 Leonhard Euler, born in Basle in 1707, died in St. Petersburg in 1783. 8 Karl Friedvich Gauss, born at Braunschweig in 1777, died at Gottingen in 1855. 4 Karl Weierstrass, born at Ostenfelde in 1815, ‘died in Berlin in 1897. The first rigorous account of the theory of real numbers which he had expounded in his lectures since 1860 was recently given by G. Mittag-Leffler, one of his pupils, in his essay: Die Zahl, Einleitung zur Theorie der analytischen Funk- tionen, The Tohoku Mathematical Journal, Vol. 17, pp. 1567—209. 1920. 5 Georg Cantor, born in St. Petersburg in 1845, died at Halle in 1918: cf. Mathem. Annalen, Vol. 5, p. 123. 1872. 8 Richard Dedekind, born at Braunschweig in 1831, died there in 1916: cf. his book: Stetigkeit und irrationale Zahlen, Braunschweig 1872. 9 Introduction. dealing with the fundamental parts of higher analysis can claim validity unless it takes the refined concept of the real number as its starting- point. ; Hence the theory of real numbers has been stated so often and in so many different ways since that time that it might seem super- fluous to give another very detailed exposition’: for in this book (at least in the later chapters) we wish to address ourselves only to those already acquainted with the elements of the differential and integral calculus. For a theory of infinite series, as will be sufficiently clear from later developments, would be up in the clouds throughout, if it were not firmly based on the system of real numbers, the only possible foundation. On account of this, and in order to leave not the slightest uncertainty as to the hypotheses on which we shall build, we shall discuss in the following pages those ideas and data from the theory of real numbers which we shall need further on. We have no intention, however, of constructing a statement of the theory compressed into smaller space but otherwise complete. We merely wish to make the main ideas, the most important questions, and the answers to them, as clear and prominent as possible. So far as the latter are concerned, our treatment throughout will certainly be detailed and without omis- sions; it is only in the cases of details of subsidiary importance, and of questions as to the completeness and uniqueness of the system of real numbers which lie outside the plan of this book, that we shall content ourselves with shorter indications. ? An account which is particularly easy to follow and which includes all the essentials is given by H.v. Mangoldt, Einfiihrung in die hohere Mathematik, Vol. I, 2m edition, Leipzig 1919. — The treatment of. G. Kowalewski, Grund- ziige der Differential- und Integralrechnung, Leipzig 1909, is accurate and concise (but for that very reason somewhat more difficult for a beginner). — A rigorous construction of the system of real numbers, which goes into the minutest details, is to be found in A. Loewy, Lehrbuch der Algebra, Part 1 Leipzig 1915, and in A. Pringsheim, Vorlesungen iiber Zahlen- und Funktionen- lehre, Vol. I, Part. I, Leipzig 1916 (cf. also the review of the latter work by H. Hahn, Gott, gel. Anzeigen 1919, pp. 321—47). Pare 1. Real numbers and sequences. Chapter'l Principles of the theory of real numbers. § 1. The system of rational numbers and its gaps. What do we mean by saying that a particular number is “known” or “given” or may be “calculated”? What does one mean by saying that he knows the value of V2 or z, or that he can calculate V5? A question like this is easier to ask than to answer. Were I to say that V2 = 1-414, I should obviously be wrong, since, on multi- plying out, 1-414><1-414 does not give 2. If I assert, with greater caution, that V2 = 1-4142135 and so on, even that is no tenable answer, and indeed in the first instance it is entirely meaningless. The question is, after all, how we are to go on, and this, without further indication, we cannot tell. Nor is the position improved by carrying the decimal further, even to hundreds of places. In this sense it may well be said that no one has ever beheld the whole of Ve, — not held it completely in his own hands, so to speak—whilst a statement that }/9 = 3 or that 35-7 = 5 has a finished and thorough- ly satisfactory appearance. The position is no better as regards the number x, or a logarithm or sine or cosine from the tables. Yet we feel certain that V2 and zz and log 5 really do have quite definite values, and even that we actually know these values. But a clear notion of what these impressions exactly amount to or imply we do not as yet possess. Let us endeavour to form such an idea. Having raised doubts as to the justification for such statements as “I know V2”, we must, to be consistent, proceed to examine how far one is justified even in asserting that he knows the number — 2% or is given (for some specific calculation) the number 2. Nay more, the significance of such statements as “I know the number 97” or “for such and such a calculation I am given a = 2andb = 5” would id Chapter I. Principles of the theory of real numbers. require scrutiny. We should have to enquire into the whole signi. ficance or concept of the natural numbers 1,2, 3, ... This last question, however, strikes us at once—and justly so—as transgressing the bounds of Mathematics and as belonging to an order of ideas quite apart from that which we propose to de- velop here. = No science rests entirely within itself: each borrows the strength of its ultimate foundations from strata above or below it, such as experience, or theory of knowledge, or logic, or metaphysics, .. Every science must accept something as simply given, and on that it may proceed to build. The only question which has to be settled by a criticism of its foundation and logical structure is what shall be assumed as in this sense “given”; or better, what minimum of initial assumptions will suffice, to serve as a basis for the subsequent development of all the rest. For the problem we are dealing with, that of constructing the system of real numbers, these preliminary investigations are tedious and trou blesome, and have actually, it must be confessed, not yet reached any entirely satisfactory conclusion at all. A discussion adequate to the present position of the subject would consequently take us far beyond the limits of the work we are contemplating. Instead, there- fore, of shouldering an obligation to assume as basis only a minimum .of hypotheses, we propose to regard at once as known (or “given”, or “secured”) a group of data whose deducibility from a smaller body of assumptions is familiar to everyone — namely the system of rational numbers, 1. e. of numbers integral and fractional, positive and nega. tive, including zero. Speaking broadly, it is a matter of common knowledge how this system may be constructed, if—as a smaller body of assumptions—only the ordered sequence of natural num- bers 1, 2, 3, ..., and their combinations by addition and multipli- cation, are regarded as “given”. For everyone knows— and we merely indicate it in passing—how fractional numbers arise from the need of inverting the process of multiplication, — negative numbers and zero from that of inverting the process of addition. The totality, or aggregate, of numbers thus obtained is called the system of rational numbers. Each of these can be completely and literally “given” or “written down” or “made known” with the help of at most two natural numbers, a dividing bar and possibly a minus sign. For brevity, we represent them by small italic characters; a, b,..., x,%,... The following are the essential properties of this system: 1 A detailed account of this type of structure will be found in the works of Loewy and Pringsheim mentioned in the Introduction; also in O. Hilder, Die Avithmetik in strengev Begriindung, Leipzig, 1914; and O. Stolz and J. A. Gmeiner Theoretische Avithmetik, 3rd edition, Leipzig 1911. : § 1. The system of rational numbers and its gaps. 5 ] t . . | 1. Rational numbers form an ordered aggregate; meaning that ‘between any two, say 4 and b, one and only one of the three relations | a=, a > 0 feswore necessarily holds; and these relations of “order” between rational numbers are subject to a set of quite simple laws, which we assume known, the only essential ones for our purposes being the Fundamental Laws of Order. x. 1. Invariably q¢ = o® 2. a =>) always implies b =a. B.a=p, D==¢ implies qg==0¢, 4. 4 #3”, or the like) and if a) this statement is correct for n= Ps and b) its correctness for wm =p, p+ 1,...,k (where k is any na- tural number > p) always implies its correctness for n = k -- 1, then the statement is correct for every natural number > $. (In applica- tions p is usually 1; it is often convenient also to take p = 0.) We will lastly mention a theorem susceptible, in the domain of rational numbers, of immediate proof, although it becomes axiomatic in character very soon after this domain is left; namely the VI. Theorem of Eudoxus. If a and b are any two positive rational numbers, then a na- tural number » always exists such that nb > a 5. The four ways of combining two rational numbers give in every case as the result another rational number. In this sense the system of rational numbers forms a closed aggregate (natiirlicher Rationalitits- bereich or mumber corpus). This property of forming a closed system with respect to the four rules is obviously not possessed by the aggregate of all natural numbers, or of all positive and negative 8 This theorem is usually, but incorrectly, ascribed to Archimedes; it is already to be found in Euclid, Elements, Book V, Def. 4. { J I | | § 1. The system of rational numbers and its gaps. r 7 integers. These are, so to speak, too sparsely sown to meet all the demands which the four rules make upon them. This closed aggregate of all rational numbers and the laws which hold in it, are then all that we regard as given, known, secured. As that type of argument which makes use of inequalities and absolute values, may be a little unfamiliar to some, its most important rules may be set down here, briefly and without proof: I. Inequalities. Here all follows from the laws of order and monotony. In particular 1. The statements in the laws of monotony are reversible; e. g. a+c¢ < b+4-¢ always implies a 0. 2. a" «=>" and “££” are substituted for “<<”. . II. Absolute values. Definition: By [a!l, the absolute value (or modulus) of a, is meant that one of the two numbets +a and —a which is positive, supposing a # 0; and the number 0, if a =0. (Hence |0|=0 and if a $0, la|> 0). The following theorems hold, amongst others: lL ja|=]|—a]l. 2. |ab|=|a]|-|b]. 1 1 b | 6] : l= Ihr provided a 4 0. 4. [a+b|Z|a|+ |b]; |a+b|=]|a]|—|b], and indeed [a+ b| Z|lal-10]|. 5. The two relations |a | <7 and —7 < 0, p=z0, both integers), is obtained by marking off on the axis, |p| times in succession, beginning at 0, the ¢th part of the distance OU (immediately constructed by elementary geometry) either in the direction OU, if $ > 0, or if pis negative, in the oppo- site direction. This point we call for brevity the point a, and the to- Ss 8 Chapter I. Principles of the theory of real numbers. tality of points corresponding in this way to all rational numbers we. shall refer to as the rational points of the axis. — The straight Line is usually thought of as drawn from left 0 right and U chosen to the right of 0. In this case, the words positive and negative obviously become equivalents of the phrases: fo the right of O and fo the left of O, respectively; and, more generally, a < b signifies that a lies to. the left of b, b to the right of a. This mode of expression may often assist us in illustrating abstract relations between numbers. This completes the sketch of what we propose to take as the previously secured foundation of our subject. We shall now regard the description of these foundations as characterizing the concept of number; in other words, we shall call any system of conceptually well-distinguished objects (elements, symbols) a number system, and its elements numbers, if — to put it quite briefly for the moment — we can operate with them in essentially the same ways as we do with rational numbers. We proceed to give this somewhat inaccurate statement a pre- cise formulation. We consider a system S of well-distinguished objects, which we denote by «, f,.... S will be called a number system and its elements , By... will be called numbers if, besides being capable of definition exclusively by means of rational numbers (i. e. ultimately by means of natural numbers alone)? these symbols «, f,... satisfy the follow- ing four conditions: 1. Between any two elements ¢ and § of S one and only one of the three relations a fy a=5, a>" necessarily holds (this is expressed briefly by saying that S is an ordered system) and these relations of order between the elements of S are subject to the same fundamental laws 1 as their analogues in the system of rational numbersS®. 2. Four distinct methods of combining any two elements of S are defined, called Addition, Subtraction, Multiplication and Division. With a single exception, to be mentioned immediately (3) these processes can always be carried out to one definite result, and obey the same 8 We shall come across actual examples in § 3 and § 5; for the moment, we may think of decimal fractions, or similar symbols constructed from ratio- nal numbers. ? Cf. also footnotes 2 and 4. 8 As to what we may call the practical meaning of these relations, nothing is implied; “ respectively). [This is also expressed, more shortly, by saying that the system S contains a sub-system S’ similar and isomorphous to the system of rational numbers. Such a sub-system is in fact constituted by those elements of S which we have associated with rational numbers]. In such a correspondence, an element of S associated with the rational number zero, and all elements equal to it, may be shortly referred to as the “zero” of the system of elements. The exception mentioned in 2. then relates to division by zero! 9 With reference to these four processes it should be noted, as in the case of the symbols ” to the “zero” of the system, there exists a natural num- ber n for which nf > «. Here nf denotes the sum g-+g-4---+f containing the element § » times. (Postulate of Eudoxus; cf. 2, VI) To this abstract characterisation of the concept of number we will append the following remark??: If the system S contains no other. elements than those corresponding to rational numbers as specified in 3, then our system does not differ in any essential feature from the system of rational numbers, but only in the (purely external) de- stgnation of the elements by symbols, or in the (purely practical) infer- pretation which we give to these symbols; differences almost as ir- relevant, at bottom, as those which occur when we write figures at one time in Arabic characters, at another, in Roman or Chinese, or take them to denote now temperature, now velocity or electric charge. Disregarding external characteristics of notation and practical inter- pretation, we should thus be perfectly justified in considering the system S as identical with the system of rational numbers and in this sense we may put g=uo,b=p5,.... If, however, the system S contains other elements besides the above mentioned, then we shall say that S includes the system of rational numbers, and is an extension of it. Whether a system of this more comprehensive kind exists at all, remains for the moment an open question; but an example will come before our notice presently in the system of real numbers. Having thus agreed as to the amount of preliminary assumption we require, we may now drop all argument on the subject, and again raise the question: What do we mean by saying that we know the number V2 or m or anything similar? from these by the process of division then form the sub-system S’ of S in question, similar and isomorphous to the system of rational numbers, as is at once clear from the construction itself. — Thus, as asserted, our concept of number is already determined by the requirements of 4,1, 2 and 4. 12 We have defined the concept of number by a set of properties charac- terizing it. A critical construction of the foundations of Arithmetic, which is quite out of the question within the limits of this volume, would have to comprise a strict investigation as to the extent to which these properties are independent of one another, i. e. whether any one of them can or cannot be deduced from the rest as a provable fact. Further, it would have to be shewn that none of these fundamental stipulations is in contradiction with any other — and other matters too would require consideration. For fuller details cf. 4. Loewy 1. c. We may note, as of particular interest, that the commutative law for addition may be deduced as a demonstrable proposition from the other fundamental laws (see Hilbert: Jahresber. d. Dtsch. Math. Ver., vol. 8, p. 1830, 1900). We could therefore omit 2, I, 8 from our fundamental laws without thereby altering the ultimate structure of Arithmetic. § 1. The system of rational numbers and its gaps. 11 It must in the first instance be termed altogether paradoxical that a number having its square equal to 2 does not exist in the system so far constructed!®, — or, in geometrical language, that the point 4 of the number-axis, whose distance from O equals the diagonal of the square of side OU, coincides with none of the “rational points”. For the rational numbers are dense, i. e. between any two of them (which are distinct) we can point out as many more as we please (since, if b—a nt1° evidently all lie between ¢ and b and are distinct from these and from one another); but they are not, as we might say, dense enough, to symbolise all conceivable points. Rather, as the aggregate of all in- tegers proved too scanty to meet the requirements of the four pro- cesses of arithmetic, so also the aggregate of all rational numbers con- tains too many gaps!? to satisfy the more exacting demands of root extraction. One feels, nevertheless, that a perfectly definite numerical value belongs to the point A and therefore to the symbol V2. What are the tangible facts which underlie this feeling? Obviously, in the first instance, this: We do, it is true, know perfectly well that the values 1-4 or 1.41 or 1-414 etc. for V2 are in- accurate, in fact that these (rational) numbers have squares < 2, i.e. are too small. But we also know that the values 1-5 or 1.42 or 1.415 etc. are in the same sense too large; that the value which we are attempting to reach would have therefore to lie between the corres- a < b, the n rational numbers given by a + » fory=1,2,..., 7, 18 Pyoof: There is certainly no natural number of square equal to 2, as 12=1 and all other integers have their squares => 4. Thus }2 could only be a (positive) fraction £ where ¢ may be taken =2 and prime to p (i. e. the 2 2, fraction is in its lowest terms). But if 2 is in its lowest terms, so is 2 = ee, which therefore cannot reduce to the whole number 2. In a slightly different . 2 : form: 2) = 2 implies p%= 24%, whence p should be an even number, say =27. p2=472=2q® however implies ¢%>=2+%, whence gq should be an even - number, in contradiction with our original assumption, that p and g were prime 2. to one another. Thus we can never have (2) = 2. This Pythagoras is said to have already known (cf. M. Cantor, Gesch. d. Mathem., Vol. 1, 27d ed., pp. 142 and 169. 1894). 14 This is the paradox, scarcely capable of any direct illustration, that a set of points, dense in the sense just explained, may already be marked on the number axis, and yet not comprise all the points of the straight line. The situation may be described thus: Integers form a first rough partition into compartments; rational numbers fill these compartments as with a fine sand, which on minute inspection inevitably still discloses gaps. To fill these will be our next problem. 12 Chapter I. Principles of the theory of real numbers. ponding too large and too small values. We thus reach the definite conviction that the value of V2 is within our grasp, although the given values are all incorrect. The root of this conviction can only lie in the fact that we have at our command a process, by which the above values may be continued as far as we please; we can, that is, form pairs of decimal fractions, with 1, 2, 3,... places of decimals, one fraction of each pair being too large, and the other too small, and the two differing ‘only by one unit in the last decimal place, i.e. by (5%)" if » is the number of decimal places. As this difference may be made as small as we please, by sufficiently increasing the number # of given decimal places, we are taught through the above process to enclose the value which we are in search of between two numbers as near as we please to one another. By a metaphor, somewhat bold at the present stage, we say that through this process V2 itself is “given”, — in virtue of it, V2 is “known”, — by it, ¥2 may be “calculated”, and so on. . : We have precisely the same situation with regard to any other value which cannot actually be denoted by a rational number, as for instance zz, log 2, sin10° etc. If we say, these numbers are known, nothing more is implied than that we know some process (in most cases an extremely laborious one) by which, as detailed in the case of V2, the desired value may be imprisoned, hemmed in, within a narrower and narrower space between rational numbers, — and this space ultimately narrowed down as much as we please. For the purpose of a somewhat more general and more accurate statement of these matters, we insert a discussion of sequences of ra- tional numbers, provisional in character, but nevertheless of fundamental importance for all that comes after. § 2. Sequences of rational numbers’. In the process indicated above for calculating Ve, successive well- defined rational numbers were constructed; their expression in decimal form was material in the description; from this form we now propose to free it, and start with the following Definition. If, by means of any suitable process of construction, we can form successively a first, a second, a third, ... (rational) num- ber and if to every positive integer m one and only one well-defined (rational) number x, thus corresponds, then the numbers CHL TER TE 1 In this section all literal symbols will continue to stand for rational numbers only. § 2. Sequences of rational numbers. 13 (vn this order, corresponding to the natural order of the integers 1, 2, 8,...,m,...) are said to form a sequence. We denote it for brevity by (2) or (m;, 2,5...) Examples. 6. iH 1 Td 1 1 T= 3 i. e. the sequence 2h or 1, FI Geen swe 9. 7, =2%; ie the sequence 2,4 3. 16 ... 3. wm, =a”; 1. e. the sequence-a, a, a%,..., where a is a given number. 1 4 a, =111— (1): 1. e the sequence 1,0, 1,0,1,0.... 5. a, = the decimal fraction for J'2, terminated at the n'd digit. 6. = erly 2 i. e. the sequence 1 In +1 Ee g yn no EE rOPLV erg or 7. Let 2,=1, 2,=1, 7,=o 1-2,=2 and, generally, for #23, let By =p q+ Typ We thus obtain the segiience 1,1, 2, 8,5, 8,18, 921,..,, usually called Fibonacci’s sequence. 1 Lis,rl 1 8. L235, =2x—%: 2, 15 =a, ces ; 3 4 5 nd 1 9. BB Igy eed —2, “oe 00 Les, 2zl Neate MY TY re y coe 11. #, =the »* prime number; i.e. the sequence 2,3, 5, 7,11, 13,..,. 11 25 137 : : = ( 1 5 12. The sequence 1, 35 10 on which z, = Ligh Ark : Remarks. ” 1. The law of formation may be quite arbitrary; it need not, in particular, be embodied in any explicit formula enabling us to obtain z,, for a given =, by direct calculation. In examples 6, 5, 7 and 11, clearly no such formula can be immediately written down. If the terms of the sequence are individually given, neither the law of formation (cf. 6, 5 and 12) nor any other kind of regularity (cf. 6, 11) among the successive numbers is necessarily apparent. 2. It is sometimes advantageous to start the sequence with a “0%” term x, or even with a (—1)® or (—2)® term, _,, x_,. We will however, even then, continue to designate as the n't term that which bears the index %#. In § 6, 2, 3 and 4, for instance, we can without further difficulties, take a 0 term or even (—1)% or (—2)t to head the sequence. The “first term” of a sequence is then not necessarily the term with which the sequence begins, The notation will be.preferably (a, #,, -..) or (x_,, @,, ...) etc., as the case may be, unless it is either quite clear or irrelevant where our enumeration begins, and the abbreviated notation (z,) can be adopted. 3. A sequence is frequently characterised as infinite. The epithet is then merely intended to emphasize the fact that every term is succeeded by other terms. It is also said that there is an infinite number of terms. More generally, there is said to be a finite number or an infinite number of things under con- sideration according as the number of these things can be indicated by a de- finite integral number or not. And we may remark here that the word infinite, when otherwise used in the sequel, will have a symbolic significance only, intended as a concise expression of some perfectly definite (and usually quite simple) circumstance. 2 Ss. 14 Chapter I. Principles of the theory of real numbers. 4. If all the terms of a sequence have one and the same value r, the sequence is said to be identically equal to c¢, and in symbols (z,)=c. More generally, we shall write (z,) = (z,/) if the two sequences (z,) and (z,) agree term for term, i. e. for every index in question z, = z,/’. 5. It is often helpful and convenient to represent a sequence graphically by marking off its terms on the number-axis, or to think of them as so marked. We thus obtain a sequence of points. But in doing this it should be borne in mind that, in a sequence, one and the same number may occur repeatedly, - even ‘infinitely often” (cf. 6, 4); the corresponding point has then to be counted (i. e. considered as a term of the sequence of points) repeatedly, or infinitely often, as the case may be. 6. A graphical representation of a different kind is obtained by marking, with respect to a pair of rectangular coordinate axes, the points whose coor- dinates are (n, z,) for n=1, 2, 3, ... and joining consecutive points by straight segments. The broken line so constructed gives a picture (diagram, or graph) of the sequence. : To consider from the most diverse points of view the sequences hereby introduced, will be the main object of the following chapters. We shall be interested more particularly in properties which hold, or are stipulated to hold, for all the terms of the sequence, or at least for all terms beyond (or, following) some definite term? With reference to this last restriction, it may sometimes be said that particular con- siderations in hand are valid “a finite number of terms being disregarded”, or only concern the ultimate behaviour of the sequence. Our first examples of considerations of the kind referred to are afforded by the following definitions: - Definitions. 1. A sequence is said fo be bounded?®, if there is a positive number K such that each term x, of the sequence satisfies . the inequality jo js R vai = RL, SR. The number K is then called a bound of the sequence. Remarks and Examples. 1. In definition 8, it is a matter of practical indifference whether we write =< KK” ‘of “= RK, For If |z, {=< K holds always (i. e. for every = in question), then we can also find a constant K’ such that |, | << K’ holds always; indeed, clearly any K’>K will serve the purpose. Conversely, if | 2, | < K always, then a fortiori |x, |< K. In particular cases, of course, the distinction may be essential. s 2. A bound K of (z,) may therefore be replaced by any larger number, _ 8. The sequences 6, 1, 4, 5, 6, 9, 10 are evidently bounded; so is 6, 3, provided |a|< 1. The sequences 6, 2, 7, 8, 11 are certainly not so. Whether 2 E. g. all the terms of the sequence 6,9 are >1. Or, all the terms of the sequence 6, 2 after the 6% are > 100 (or more shortly: for »>6, x, > 100). : 3 This nomenclature appears to have been introduced by C. jordan, Cours d’analyse, vol. 1, p. 22. Paris 1893. § 2. Sequences of rational numbers. - 15 6, 3 for every bal, or 6, 12, is bounded or not, is not immediately obvious. 4. If all we know is the existence of a constant K,, such that z, K, always, then (z,) is said to be bounded on the left (or beter) and K, is called a bound below (or a left hand bound) of the sequence. Here XK, and K, need not be positive. 5. Supposing a given sequence is bounded on the right, it may still happen that among its numbers none is the greatest. For instance, 6, 10 is bounded on the right, yet every term of this sequence is exceeded by all that follow it, and none can be the greatest? Similarly, a sequence bounded on the left need contain no least term; cf. 6, 1 and 9. — (With this fact, which will appear at first sight paradoxical, the beginner should make himself tho- roughly familiar). > Among a finite number of values these is of course always both a greatest and a least, i. e. a value not exceeded by any of the others, and one which none of the others fall below. (ieee may, however, be several equal to this greatest or least value.) I. A sequence is said to be monotone ascending or increasing 9. if, for every value of n, Zn = Zpt15 it is said to be monotone descending or decreasing if, for every n, x, > ntl Both kinds will also be referred to as monotone sequences. Remarks and Examples. 1. A sequence need not of course be either monotone increasing, or monotone decreasing; cf. 6, 4, 6, 8. Monotone sequences are, however, extremely common, and usually easier to deal with than those which are not monotone. That is why it is convenient to give them a distinguishing name. 2. Instead of “ascending” we should more strictly say “non-descending”’, and instead of “descending”, “non-ascending”. This, however, is not customary. If in any special instance the sign of equality is excluded, so that x, << xp+1 or x, > %, 11, as the case may be, for every », then the sequence is said to be strictly monotone (increasing or decreasing). ‘ 8. The sequences 6, 2, 5, 7, -10, 11, 12 and 6, 1, 9 are monotone; the first-named ascending, the others descending. 6, 3 is monotone descending, if 0 1; for a <0, it is not monotone. 4. The designation of “monotone” is due to C. Neumann (Uber die nach Kreis-, Kugel- und Zylinderfunktionen fortschreitenden Entwickelungen, pp. 26, 27. Leipzig 1881). 4 The beginner should guard against modes of expression such as these, which may often be heard: “for # infinitely large, x, = 1"; “l is the greatest number of the sequence”. Anything of this sort is sheer nonsense (cf. on this point 7, 3). For the terms of the sequence are 0, 1,2, 2,... and none of these is = 1, on the contrary all of them are <1. And there is no such thing as an “infinitely large »”. 0 Chapter I. Principles of ‘the theory of real numbers, We now come to a definition to which the reader should pay the greatest attention, sparing no effort to make himself master of its meaning and all that it implies. II. A sequence will be called a null sequence if it possesses the following property: given any arbitrary positive number &, the equality lo Jub n is satisfied by all the terms, with the exception of at most a finite number of them. In other words: an arbitrary positive number & being chosen, a number ny, can always be found, such that je 0. On this account, it is usual to formulate the definition some- what more emphatically as follows: (z,) is a null sequence if, fo every ¢ > 0, however small, there corresponds a number n, such that Yon | <= 2 for every = > u,. 2. The sequence 6, 1 is clearly a null sequence; for . : ; |, | < &, provided n> =, whatever be the value of &. It is thus sufficient to put n,=—. & 3. The place in a given sequence beyond which the terms remain nu- merically < e, will naturally depend in general on the magnitude of ¢; speaking broadly, it will lie further and further to the right (i. e. n, will be larger and larger), the smaller the given eis (cf. 2). This dependence of the number #, on ¢ is often emphasised by saying explicitly: “To each given & corresponds a number n, = #,(¢) such that...” 4. n, need not be an integer. 5. The sign of z, plays no part here, since |—a, | =|2,| and consequently both or neither << &. Accordingly 6, 6 is also a null sequence. 6. In a null sequence, no term need be equal fo zero. But all terms whose index is very large, must be very small. For if I choose ¢ =10-5, say, then for every # > a certain #n,, |x, | must be <10~% Similarly for = 10-10 and for any other e. 7. The sequence (a™) specified in 6, 3 is also a null sequence provided |a | <1. Proof. If a=0, the assertion is trivial, since then, for every > 0, 1 | 2 | < & for every n. If 0<[a]|<]1, then (by 3,1, 4) =~ 1. If there fore we put =1+2, then p>0. a § 2. Sequences of rational numbers. 17 But in that case, for every » => 2, we have (a) ! +p)" >1+np. For when #n =2, we have (14+ p)2=1+4+2p 4 2p2 > 1 2p; the stated relation therefore holds in that case. If, for n=%=>2, (1+pF>1+kp, then by 2, Ill, 6 40 > AA) =1 +E D php? > 1+(k+1)p, therefore our relation, assumed true for n» =k, is true for n=%k+1. By 2, V it therefore holds for every n= 259). : Accordingly, we now have tml =a |=]ap =p n cp loo z : Oar" taro ng’ so that, however small ¢ > 0 may be, we have op =a? 22 for every ">on 8 . : 1 ; : 1 1 8. In particular, besides the sequence = mentioned in 2, 3) \3) } EY etc. are also null sequences oT) Ts 9. A similar remark to that of 8, 1 may be appended to Definition 10: no essential modification is produced by reading “< &” for “mn,, |z,| #,, then choosing any positive number &, mn,, and consequently | mpl n,’ and “=n,” are practically interchangeable alternatives. In any individual case, however, the distinction must of course be taken into account. 10. Although in a sequence every term stands entirely by itself, with a definite fixed value, and is not necessarily in any particular relation with the preceding or following terms, yet it is quite customary to ascribe “to the terms z,”’, or “to the general term” any peculiarities in the sequence which may be observed on running through it. We might say, for instance, in 6, 1 the terms diminish; in 6, 2 the terms increase; in 6, 4 or 6, 6 the terms oscillate; in 6, 11 the general term cannot be expressed by a formula; and so on. — In this sense, the characteristic behaviour of a null sequence may be described by saying that the terms become arbitrarily small, or infini- tely small®; by which neither more nor less is meant than is contained in 5 The proof shows moreover that (a) is valid for n = 2 provided only 1+ p >, i. e.p>—1, but 0. For p=0 and for w=1 (a) becomes an equa- lity. — The relation (a) is called Bernoulli's Inequality (James Bernoulli, Prop. arithm. de seriebus, 1689, Prop. 4). - 6 This mode ‘of expression is dune to 4. L. Caucliy (Anal. alg., pp. 4 and 25 18 . Chapter I. Principles of the theory of real numbers. Definition 10, viz. that for every &> 0 however small the terms are ultimately (i. e. for all indices » >> a suitable #,; or from and after, or beyond, a certain ng) numerically less than ¢ “ 11. A null sequence is ipso facto bounded. For if we choose ¢=1, then there must be an integer =, such that, for every n>mn,, |2,|>0, the corresponding #, can actually be determined. Conversely, if a sequence (z,) is assumed to be a null sequence, it is thereby assumed possible to actually determine, for every &, the corre- sponding #,. On the other hand the student should make sure that he under- stands clearly what is meant by a sequence %ot being a null sequence. The meaning is this: it is not true that, for every positive number &, beyond a cer- tain point | z, | is always < ¢; there exists a special positive number g,, such that |z, | is not, beyond any n,, always << g,; after*every =, there is a larger index # (and therefore an infinite number of such indices) for which |x, | = ¢;. 13. Finally we may indicate a means of interpreting geometrically the special character of a null sequence. Using the graphical representation 7, 5, the sequence is a null sequence if its terms ultimately (for »>m,) all belong to the inferval® —e... +e. Let us call such an interval for brevity an e-neighbourhood of the origin; then we may state: (z,) is a null sequence if every e-neighbourhood of the origin (however small) contains all but a finite number, at most, of the terms of the sequence. Similarly, using the graphical representation 7, 6, we can state: (z,) is a null sequence if the whole of its graph, with the exception, at most, of a finite initial portion, lies in every e-strip (however narrow) about the axis of abscissae; (the latter being defined by drawing through the two points (0, 4 ¢), parallels to the axis of abscissae). The concept of a null sequence, the “arbi- trarily small given positive number &”, to which we shall from now on have continually and indispensably to appeal, and which may thus be said to form a main support for the whole superstructure of analysis, appears to have been first used in 1655 by J. Wallis (v. Opera L, p. 382/3). Substantially, however, it is already to be found in Euclid, Elements V. : We are already in a better position to comprehend what is involved in the idea, discussed above, of a meaning for Y2 or zorloghs — In forming on the one hand (we keep to the instance of V2) the numbers =14; m=141; a,=144; 9, =14142;,,., on the other, the numbers y, =158; y,=142; 9,=1415; y,=14143;... ~ 7 There need of course be no question here of the sequence being mono- tone. Also, in any case, some |x, |'s of index 0 be chosen. The number s=1 here belongs to all the J's, since = 13 1S for every #. No number other than 1 can belong fhierelore to all tie intervals. 9. Let J, be defined as follows: [;!° is the interval 0... 1; J, the left half of J,; J, the #ight half of J; Js the left half of J,; and so on. These inter- vals are obviously each contained in the preceding; and since J, has length dp = gs and these numbers form a null sequence, we have a nest of intervals. A little consideration shows that the sequence of the z,’s consists of the numbers ‘ : 1 1 1 21 1 1 IIT ITTniteTw 0, + We note here for future reference that this theorem continues to hold unaltered when the numbers which occur are arbitrary real numbers. 9 From a graphical point of view, what the proof indicates is that if s and §’ belong to all the intervals, then each interval has a length at least equal to the distance |s—s’| between s and s’ (v. 3,1, 6); these lengths cannot, therefore, form a null sequence. 10 Here we let the index start from 0 (cf. 7, 2). 7 | | | | § 2. Sequences of rational numbers. ior each taken twice running; and that the sequence of y,’s begins with 1 and continues with . gl Lg gt pal YL ay each taken twice running. Now 1... a wih BD 1 Tutat tpg a certain #,. 2% 29 Chapter I. Principles of the theory of real numbers. -and y,’=19,2>2 for all u (because this was how =z, and y, were chosen) i.e. x,’ <2 2 and accordingly put 2, =1, y,=2. We then divide the inter- val J,=x,...%, into 10 equal parts, and taking the points of division, | 28 for k=0,1,2,...,9, 10, determine by trial whether their squares are >2 or <2 We find that the squares corresponding to 2=0,1, 2, 3, 4 are too small, those corresponding to £=35, 6, ..., 10 too large, and accordingly we put x, = 1.4 and y, = 1.5. Next, we divide the interval Jy =a, ...y, into 10 equal parts, and go through a similar test with regard to the new points of division — and so on. The known process for extracting the square root of 2 is intended mainly to make the successive trials as mechanical as possible. -— § 3. Irrational numbers. 23 symbols) are equivalent, and may to a certain extent be considered equal, so that we may write indeed: (J.)=s or (x, |v.) =s. Consequently, we will not say merely: “the nest (J) defines the number s” but rather “(J ) is only another symbol for the number s”, or in fine, “(J ) 4s the number s” — exactly as we are used to look upon the decimal fraction 0-333... as merely another symbol for the number 3, or as being precisely the number % itself. It now becomes extremely natural to introduce tentatively an analogous mode of expression with regard to those nests of intervals which contain no rational number. Thus if 2, y, denote the numbers constructed previously in connection with the equation 2? = 2, one might — seeing that in the system of rational numbers there is not a single one whose square = 2 — decide to say that this nest (z, |y, determines the “true” “value of }/2” though one incapable of being symbolised by means of rational numbers, — that it encloses this value unambiguously — in fine, “it is a newly created symbol for this number”, or, for brevity, “it is the number itself”. And similarly in every other case. If (J )= (z,|y,) is any nest of intervals and no rational number s belongs to all its intervals, we might finally resolve to say that this nest encloses a perfectly definite value, — though one incapable of being directly symbolised by means of rational numbers, — it determines a perfectly definite number, — though one unfortunately non-existent in the system of rational numbers, — it is a newly created symbol for this number, or briefly: is the number itself; and this number, in contradistinction to the rational numbers, would then have to be called an irrational number. Here certainly the question arises: Can this be done without further justification? Is it allowable? May we, without more ado, designate these new symbols, the nests (x,|v,), as numbers? The following considerations are intended to show that to this course there is no obstacle whatever. The corresponding treatment of, for instance, the equation 107 =2 (i. e. deter- mination of the common logarithm of 2), involves the following nest of inter- vals: Since 10° <2, 10'>2, we here put z,=0, ¥,=1 and divide J, =a... 9, into 10 equal parts. For the points of division, 10° we next test, whether Lz 101 <2 or > 9, that is to say, whether 1042 <2 or 210, As a result of this trial, we shall have to put » =0.3, y, =0-4. The interval J, ==, ...9, is again divided into 10 equal parts, the same procedure instituted for the k points of division Sid 100 and, in consequence, x, put equal to 0-30 and y, to 0-31 — and so on. — This obvious procedure is of course much too laborious for practical calculations. I \ be exactly one, say pts such that P lies between x, = p |- ~ 24 Chapter I. Principles of the theory of real numbers. In the first instance, a simple graphical illustration of these facts on the number-axis (see fig. 1) gives every appearance of justification to our resolution. If, by any construction, we have marked a point P | on the number-axis (e. g. by marking off to the right of O the length of the diagonal of a square of side OU) then we can in any number of ways define a nest of intervals enclosing the point P. We may do so in this way, for instance. First of all we imagine all integers £0 marked on the axis. Of these, there will be exactly one, say p, such that our point P lies in the stretch from $ inclusive to (p +4 1) exclusive. Accordingly we put z,=p, y,=p-+1 and divide the 6 o ss vn ply ef 1 7 1 7 2 7 = = —— J, Sg Pig. 1. interval J,=x,...y, into 10 equal partsf, The points of division are p -1- > {wih £2=0,12..}, 10), and among them, there will again ky 10 541 10 is again divided into 10 equal parts, and so on. If we imagine this process continued indefinitely, we obtain a perfectly definite nest (J) all of whose intervals J contain the point P. No other point P’ be- sides P can lie in all the intervals J,. For, if that were so, all the intervals would have to contain the whole stretch PP’, which is 1 10% inclusive and y, =p | exclusive. The interval J =z, ... 9, i impossible, as the lengths of the intervals ( J, has length ) form a null sequence. For every arbitrarily given point P on the number-axis (rational or not) there are thus nests of intervals — obviously, indeed, any number of such nests — which contain that point and no other. And in the present instance, — i. e. in the graphical representation on the number-axis —, the converse appears most plausible; if we consider any nest of intervals, there seems to be always one point (and by the reasoning above, only this one) belonging to all its intervals, which is thus determined by it. We believe, at any rate, 14 Instead of 10 we may of course take any other integer => 2. For further detail, see § 5. § 3. Irrational numbers. : 25 that we may infer this directly from our conception of the continuity, or gaplessness, of the straight line. Thus in this geometrical representation we should have complete reciprocity: every point can be enclosed in a suitable nest of inter- vals and every such nest invariably encloses one and only one point. ' The idea of the so-called continuity of a straight line is however undoubtedly vague, scarcely capable of being conceptually defined with entire precision; it may not therefore, in the construction of analysis, be taken — and the same is true of all other ideas derived from intuitive illustrations — to form part itself of the ultimate foundations, but only used at most to point out the direction in which the framing of the latter should proceed. Consequently we are not permitted to translate the geometrical considerations just brought for- ward into a satisfactory theorem; but we derive from them a high degree of confidence in the adequacy and justification of our reso- * lution — namely to consider nests of intervals as numbers, — which we now formulate more precisely as follows: Definition. We will say of every mest of intervals (J,) or (x,|y,), 18. - that it defines, or, for brevity, it is, a determinate number. To re- present it, we use the symbol denoting the mest of intervals itself, and only as an abbreviation replace this by a small Greek letter, writing in this sense e. g. . (Jn) or (xn|yn)=0."° Now, in spite of all we have said, this cannot but seem a very arbitrary step,-— the question has to be repeated most insistently: will it pass without further justification? These purely ideal objects which we have just defined — these nests of intervals (or else that still extremely questionable ‘something’ which such a nest encloses or determines) can we speak of these as numbers? Are they after all numbers in the same sense as the rational numbers, — more preci- sely, in the sense in which the number concept was defined by our conditions 47? : The answer can only consist in deciding, whether the totality or aggregate of all conceivable nests of Intervals, or of the symbols (J ) or (x, |v,) or o introduced to denote ‘them, forms a system of objects satisfying these conditions 417; a system therefore — to recapitulate 15 The proposition, by which the “continuity of the straight line” is ex- pressly postulated — for a proof cannot be here expected, since it is essentially a description of the form of our concept of the straight line which is involved — is called the Cantor-Dedekind axiom. 1% I. e. ¢ is an abbreviated notation for the nest of intervals (J,) or (tn | | Yn): 17 The reader should here read these conditions through again. 26 Chapter I. Principles of the theory of real numbers. these conditions briefly — whose elements are derived from the rational numbers, and 1. are capable of being ordered; 2. are capable of being combined by the four processes (rules), obeying at the same time the fundamental laws 1 and 2, I—1V; 3. contain a sub-system similar and isomorphous to the system of rational numbers; and 4. satisfy the Postulate of Eudoxus. If and only if the decision turns out to be favourable, all will be well; our new symbols will then have vindicated their numerical | character, and we shall have established that they are numbers, - whose totality we shall then designate as the system of real numbers. Now the decision in question does not present the slightest difficulty, and we may accordingly be brief in expounding the details: Nests of intervals — or our new symbols (x |v) — are cer tainly constructed by means of rational number-symbols alone; we have therefore only to settle the points 4, 1—4. For this, we shall go to work in the following way: Certain of the nests of inter vals define a rational number 1%, something, therefore, for which both meaning and mode of combination have been previously established. We consider two such rational-valued nests, say (z,|y,)=s and (x.|y,)=s". With the two rational number-symbols s and s’, we can immediately distinguish whether the first s is <, = or > the second §’; and we can combine the two by the four processes of arithmetic. Essentially, what we have to do is to endeavour directly to recognise the former fact, and to carry out the latter processes, on the two nests of intervals themselves by which s and s" were given, and finally to “extend the result to the aggregate of all mests of intervals. Each 14. provable proposition (A) relating to rational valued nests will accordingly give rise to a corresponding definition (B). We begin by setting down concisely side by side these pairs of propositions (4) and definitions (B).2? Equality: A. Theorem. If (x, |y,) =s and (x, |y,))= +s" are two rational-valued nests of intervals then s=s holds if and only if, besides 2 ’, i. e. s—s’ > 0, then, since (y,—x,) is a null sequence, we could so choose the index p, that Vp — Bp 5—4. As however s is certainly =< y,, this would imply z, —s'>0. We could there- fore choose a further index » for which Vv =u) << wy —o, Since 2,’ < ¢’, this would imply y,’ << z,. Choosing an integer m exceed- ing both p and 7, we could deduce, in view of the respective ascending and descending monotony of our sequences of numbers, that a fortiori vy,’ < x, — which contradicts the hypothesis that x, 0, 1. e. “positive”, if and only if (wm, |v, > (0, 0), that is to say, if for some suitable index p, x, > 0. But in this case, as the w,’s increase with », we have a fortiori x, > 0 for every n> p. We may there- fore say: o = (,|y.) is positive if, and only if, all the endpoints x,, y, are positive from and after a definite index. — The exact analogue holds of course for ¢ < 0. ~ 5. If then ¢ 0, and, for every n= p, u, > 0, let us form a new nest &l 9.) =0 by patting a) =n)l=.-.= =a ; all equal to z,, but every other z,’ and y,/ equal to the corresponding x, and y,. By 14, obviously ¢ =o’; and we may say: If ¢ is positive, then there are always nests of intervals equal to it, for which all the endpoints of intervals are positive. The exact analogue holds for o <0. So far then, in respect of the possibility of ordering them, our nests of intervals may be said to vindicate their character as numbers completely. It is no more difficult to establish a similar conclusion with regard to the possibilities of combining them. Addition: A. Theorem. If (x, |y,) and (z,|y,) are any two nests of intervals, then (x,4-z,, v,+y,’) is also one, and if the former are both rational-valued and respectively =s and =3s', then the latter is also rational-valued, and determines the number s Fs’ ?3, B. Definition. If (x, |y,)=0 and (x, |y,))=d are any two nests of intervals and o” denotes the mest (x, +x’, v, + y,) deduced from them, then we write BS and o' is called the sum of o and do. Subtraction: A. Theorem. If (v,|y,) is a nest of intervals then so is (—y,| —=,); and if the former is rational-valued ='s, then the latter is also rational-valued, and determines the number — s. B. Definition. If o=(x,|y,) is any nest of intervals and d de- note the mest of intervals (— vy, | —x,), we write d= —o and say o is the opposite of 6. — By the difference of two nests of inter- vals we then mean the sum of the first and of the opposite of the second. 23 With regard to the proof, cf. footnote 20. ~ 16. 1%7. 18. 19. - w= 30 Chapter I. Principles of the theory of real numbers. Multiplication: A. Theorem. If (x, |y,) and (x,|y,’) are any two positive mests of intervals, — replaced, if necessary, (in accordance 1 with 15, 5) by two nests of intervals equal to them, for which all the endpoints of intervals are positive (or at least non-negative), — then («|v v,) is also a nest of intervals; and if the former are rational- valued and respectively = s and =’, then the latter is also rational- valued, and determines the number ss’. B. Definition. If (z,|y,) =o and (x, |y,)) = 0 are any two po- sitive nests of intervals for which all the endpoints of intervals are positive — which is no restriction, by 15,5 — and d’ denote the nest (x, x, |y,5,) derived from them, then we write > oc’ = Gd and call o” the product of ¢ and o'. The slight modifications which have to be made in this definition if one or both of ¢ and ¢ are negative or zero, we leave to the reader, and henceforth consider the product of any two nests of intervals as defined. Division: A. Theorem. If (x, |v,) is any positive nest of intervals for which all endpoints of intervals are positive, (cf. 15, 5) then so is (15): and if the former is rational-valued, and =s, the latter is also rational-valued, and determines the number 2 B. Definition. If (2, |y,)= 0 is any positive nest of intervals for which all endpoints are positive, and o denote the nest = 2) , then we write 1 ’ 0 = — o and say o is the reciprocal of 6. — By the quotient of a first by a second positive nest of intervals we then mean the product of the first by the reciprocal of the second. ~The slight modifications necessary in this definition, if ¢ (in the one case) or the second of the two nests of intervals (in the other) is negative, we may again leave to the reader, and henceforth consider the quotient of any two nests of intervals of which the second is different from 0, as defined. — If (x |y,)=0=0, then the above method fails to produce a “reciprocal” nest: division by 0 is here also impossible. The result of the preceding considerations is thus as follows: By definitions 14 to 19, the system of all nests of intervals is ordered in the sense of 4,1, and admits of having its elements combined by the four processes in the sense of 4, 2. In consequence of the theo- rems 14 to 19, as stated in each case, this system possesses further, 8 3. Irrational numbers. : 31 in the aggregate of all rational-valued nests, a sub-system, similar and isomorphous to the system of rational numbers, in the sense of 4, 3. It remains to show that the system also fulfils the Postulate of Eudoxus. But if (x,|y,)=o0 and (z,'|y,/)= 0’ are any two positive nests for ~which all endpoints of intervals are positive (cf. 15, 5), let «and y,, be a definite pair of these endpoints; the theorem of Eudoxus ensures the existence of an integer p, for which pm, => yp, and the nestpo, or (px,|py,), in accordance with 15, is then effectually > ¢'. . The next step should be to establish in all detail (cf. 14, 4 and 15, 2) that the four processes defined in 16 to 19 for nests of inter- vals obey the fundamental laws 2. This again offers not the slightest difficulty and we will accordingly spare ourselves the trouble of setting it forth®t. The Fundamental Laws of Arithmetic, and thereby the entire body of rules valid in calculations with rational numbers, effectually retain their validity in the new system. By this, our nests of intervals have finally proved themselves in every respect to be numbers in the sense of 4: The system of all nests of intervals is a number-system®®, the nests themselves are numbers®S. 2 As regards addition, for instance, it should be shown that: a) Addition can always be carried out. (This follows at once from the definition.) b) The result is unique; i.e. =o’, v=7 (in the sence of 14) imply 0+ t=0"+1, — if the sums are formed in accordance with 16 and the test for equality carried out in accordance with 14. In the corresponding sense, it should be shown further that c) 6 +7 =17-+o0 always. d) (+0)+7=0+ (047) always. €) 6 < ¢’ implies 6 +7 << 0’ +7 always. — And similarly for the other three processes of combination. 2 The mode of defining the number-concept given in 4 is of course not the only possible one. Frequently the designation of number is still ascribed to objects which fail to satisfy some one or other of the requirements there laid down. Thus for instance we may relinquish the condition that the objects under consideration should be constructively developed from rational numbers, regarding any entities (for instance points, or distances, or such like) as num- bers, provided only they satisfy the conditions 4, 1—4, or, in short, are similar and isomorphous to the system we have just set up. — This conception of the notion of number, in accordance with which all isomorphous systems must be regarded as in the abstract sense identical, is perfectly justified from a mathe- matical point of view, but objections necessarily arise in connection with the theory of knowledge. — We shall encounter another modification of the num- ber-concept when we come to deal with complex numbers. = 26. Whether, as above, we regard nests of intervals as themselves num- bers, or imagine some hypothetical entity introduced, which belongs to all the intervals J, (cf. 15, 8) and thus appears to be in a special sense the number enclosed by the nest of intervals and, consequently, the common element in all equal nests — this at bottom is a pure matter of taste and makes no essential * 392 Chapter I. Principles of the theory of real numbers. This system we shall henceforth designate as the system of real numbers. It is an extension of the system of rational numbers, — in the sense in which the expression was used on p. 10, — since there are not only rational-valued nests but also others besides. = This system of real numbers is in one-one correspondence with the whole aggregate of points of the number-axis. For, on the strength of the considerations set forth on pp. 23, 24, we can immediately assert that to every nest of intervals ¢ corresponds one and only one point, namely that common to all the intervals J , which on account of the Cantor-Dedekind axiom is considered in each case as existing. Also two nests of intervals ¢ and ¢ have, corresponding to them, one and the same point, if and only if they are equal, in the sense of 14. To each number ¢ (that is to say, to all nests of intervals equal to each other) corresponds exactly one point, and to each point exactly one number. The point corresponding in this manner to a particular number is called its image (or representative) point, and we may now assert that the system of real numbers can be uniquely and reversibly represented by the points of a straight line. '§ 4. Completeness and uniqueness of the system of real numbers. Two last doubts remain to be dispelled®?: Our starting point in § 3 was the fact that the system of rational numbers, by reason of its “gaps”, could not satisfy all demands which would appear in the course of the elementary processes of calculation. Our newly created number-system, — the system Z as we will call it for brevity — is in this respect certainly more efficient. E. g. it contains a number ¢ for which ¢2 = 22. Yet the possibility is not excluded that the new system may still show gaps like the old, or that in some other way it may be susceptible of still further extension. Accordingly, we raise the following question: Is it conceivable that a system Z, recognizable as a number system in the sense of 4, and containing all the elements of the system Z, should also contain additional elements distinct from these? *® difference. — The equality ¢ = (2, |¥,) we may, at any rate, from now on, (cf. 13, footnote 16) read indifferently either-as “sc is an abbreviated notation for the nest of intervals (a | 2)”y or as “c is the number defined by ais nest of intervals (x, |¥,)"- 27 Cf, the closing words of the Introduction (p. 2). - 28 For if 6 = (2,!| ¥,) denote the nest of intervals constructed on pp. 18, 19 in connection with the Souetion 22 = 2, then by IS we have I ot | ¥2%). Since, however, x,2 <2 and ¥,% > 2, it follows that ¢2=2. Q.E 2 i e. Z would have to represent an extension of Z in the same sense as Z itself represents an extension of the system of rational numbers. ~ § 4. Completeness and uniqueness of the system of real numbers, 33 It is not difficult to see that this cannot be so, so that we have in fact the following theorem: Theorem of completeness. The system Z of all real numbers 1s incapable of further extension compatible with the conditions 4. Proof: Let Z be a system which satisfies the conditions 4 and contains’ all the elements of Z. Uf « denote an arbiirary: element of 2. then 4, 4, — in which we choose for § the number 1, contained in Z, and also, therefore, in Z, — shows that there exists an integer p > «, and similarly another p’ > — «. For these we have — p' < nor the non-negative one of the two elements 8 — 8’ and 3’ — 3. Let 7 stand for an arbitrary positive rational number, and r for the corresponding element in 3 (therefore in J); then, on account of the similarity and isomorphism of 3’ with the system of rational numbers, we must have, simultaneously with ¥, eC 7, the relation Ie oe holding for a suitable index $. For every such r therefore 8 —8| 2, we have the exact analogue for fractions in a scale of radix g or radix fractions with base g. To begin with, given a real number ¢, an integer p (>,=, or <0) is uniquely defined by the condition p2 — will prove periodic (or recurring) if and only if 6 is rational®. : A particularly advantageous choice to make is often g= 2; the process for expressing the number ¢ is then called briefly the method of bisection and the resulting radix fraction, whose digits can in that case only be 0 or 1, is called a binary fraction. The method, in a somewhat more general light, is this: we start from a definite interval J; and, in accordance with some particular rule or point of view, definitely select one of its two halves, calling it J; we then again make a definite choice of one of the two halves of J, calling it J,; and so on. By so doing, we specify, in every case, a well defined real number, determined with absolute uniqueness by the method which regulates at each stage the choice between the two half-intervals 3°. 33) That we have a nest of intervals is immediately obvious, since 1 Tp S Tp < Yn = Yn—y throughout, and Turn =r forms a null sequence, by 10, 7. ~ 3 The slight alteration in our method, required if all the intervals are considered as including their right and not their left endpoints, the reader will doubtless be able to carry out for himself. The two results differ if, and only if, the given number o is rational, and can be written as a fraction having, as denominator, a power of g, so that the point ¢ is an endpoint of one of our intervals. — Actually, the two nests of intervals p+0-24...2. (7, — 1) (g—1D)(g—1)... and pt 0.22...2.,2,00... are equal by 14, where the digit 2. is supposed == 1. In every other case, two radix fractions which are not identical are unequal, by 14. — The reader will easily prove for himself that, except in this case, the representation of any real number ¢ as a radix fraction with base g is absolutely unique. 3% Here for simplicity we regard terminating radix fractions as periodic with period 0. — That every rational number can be represented by a recur- ring decimal fraction was proved by J. Wallis, De Algebra tractatus p. 364, 1693. That conversely every irrational number can always, and in one way only, be represented as a non-recurring decimal fraction was {first proved generally by O. Stolz (v. Allgemeine Arithmetik I, p. 119, 1885). 3 An example was given in 12, 2. 33 Chapter I. Principles of the theory of real numbers. : In radix fractions, just as in decimal fractions, we accordingly see a peculiarly clear and convenient mode of specifying nests of inter- vals. They shall accordingly in future be admitted for the definition of real numbers on the same footing as decimal fractions. The distinction lies somewhat deeper between nests of intervals and the following method of definition of real numbers. Fo We suppose given, in any particular way, two classes of numbers A and B??, subject to the following three conditions: 1) Each of the two classes contains af least ome number. 2) Every number of the class 4 is < every number of the class B. 3) If an arbitrary positive (small) number ¢ is prescribed, then two numbers can be so chosen from the two classes, — 4/, say, from 4 and ¥, say, from B,~— that b — a’ < &38, '— Then the following theorem holds: Theorem 3. There exists one and only one real number oc such that for every number a in A and every number b in B the relation ga¢ for every pair of elements ¢ and b from 4 and B re spectively. There exists then at most one such number 6. We find it in the following way: By hypothesis, there is at least ome number a, in 4 and one number b, in B. If a, =b,, then the common value is manifestly the number ¢ which we are in search of. If a, &=b,, and therefore by 2), a, b, and apply the method of bisection to the interval J, which they determine; we denote the left or right half by J,» according as the left half (endpoints included) does or does not still contain a point of the class B. By the same rule we next select one of the halves of J,, calling it J;, and so on. 37 E. g. 4 contains all rational numbers whose cube is << 5, B all ratio- nal numbers whose cube is >5, — or the like. 38 We say for short: the numbers of the two classes approach arbitrarily near to one another. In the example of the preceding footnote, we see at once that conditions 1) and 2) are satisfied; that 3) is also satisfied we recognise from the possibility of calculating (by the method of partition into tenth parts, for instance) two decimal fractions x, and y, with » places of decimals, differing only by a unit in the last place, and such that z,> <5, 3,°> 5; n being so 1 chosen that Tor <8, § 5. Radix fractions and the Dedekind section. 39 The intervals J, J,,..+ J ;..., being obtained bythe method of bisection, necessarily form a nest From their mode of formation, they possess moreover the property that no number of B can lie to the left of any of their left endpoints, and no number of 4 to the right of their right endpoints. But from this it follows at once that the number ¢ enclosed by them is the number required by theorem 3. In fact, if, contrary to the assertion in that theorem, a particular number g of 4 were > o, so that a — 6 > 0, then we could choose from the succession of intervals J a particular one, say 5 = voy Yor with length < a — 6. Since z, <0 Jn 7° ’ (r=1,2...); db) 0 = zy <= Vy and for eyery =» = 1, Cntr = Yo. Yas Yn = 1 (2, =f Va) c) 0 iY on ” ” Nn Upp = $ (z, = Vn)» Yat+1 = Vooriy Vn y d) 0 << n » ” no vary (@, + 7) Cntr = Von Vass y e) 0 BV » ” ” ny Typ = Vz, Vas Yut1 = 1 (0g ts Vn) > f) 0 << ow; Yi ” ” By Pp iy = Vou Yny Tongs = % (Zn + Yuta) ’ Tn+y D0<5, ,, for every n. All remarks made in 8 and 9 retain their full validity. Examples: 1. The sequences 23, 1, 2, 4 and 5 are evidently bounded. . . . . : . | 2. A sequence (x,) is said to be monotone increasing if x, m, and accordingly 1 Tn Terms of the sequence may therefore always be found, which b are > K or <<— K, however large the constant K is chosen. — For 5. the boundedness follows from the fact that all the terms lie between a and b. 2. The sequences 23, 1 and 2 are monotone decreasing: the others are not monotone. The definition 10 of a null sequence and the appended remarks — which the student should read through again carefully — also remain unchanged. © Definition. A sequence (x) shall be termed a mull sequence if, 25. subsequently to the choice of am arbitrary positive number &, a number ny = ny (€) may always be assigned, such that the inequality jz, | n,. Examples. 1. The sequence 28, 1 is a null sequence, for the proof 10, 7 is valid for any real a, for which |a| <1. 2. 23, 2 is also a null sequence, for here onl s) therefore 0 be given. By the assumption, we can assign n,, so that for every n > n,, 8 7, l< 5 ( But for the same # we then always have |2) | =la,| |, | m and e¢ > 0 is given, then by the assumptions we can assign un, > m, so that for every un > #,, || < >. Since for these values of # we then also have || n,, |2,| < ¢, then we have, ipso facto, for any such #, jo) =(n, le, since k, is certainly > n,, when = is. OTheorem 2. Let an arbitrary sequence (x,) be separated into two sub-sequences (x) and (x,”), — so that, therefore, every term of (x) belongs to one and only ome of these sub sequences. If (x) and (x) are both mull sequences, then so is (x,) itself. : 8 This “may” of course be done without any particular permission. This mode of expression is only meant to convey that in so doing we do not disturb the particular property of the sequence which interests us at the mo- ment, viz. the property of being a null sequence. SIR 0 be chosen, then by hypothesis a num- ber #' exists, such that for every n > #’, || #", |x" | n,, obviously |z | n,, |2,| <ée. Among the indices belong- ing to the finite number of places which the terms #,, wm, ...., 2, ~ 0 occupy in the sequence (z,’), let #’ be the largest. Then obviously, for every m > #/, |2,’| < &; hence (x) is also a null sequence. + OTheorem 4. If (x,) is a null sequence and (x,') is obtained from it by any finite number of allerations®, then (z,') is also a null se quence. The proof follows immediately from the fact, that for a suitable integer p20, from some » onwards we must have x’ =a ,, For if every x, for » > n, has remained unchanged, and x,, has received the index #' in the sequence (z,’), then in point of fact for every n>, 2, == ty if we put p=mn, —n'". Theorem 5. If (x) and (x) are two null sequences and if the n sequence (x,) is so related to them that from a certain m onwards 2/ Lo, Saf (n> m) then (x,) is also a null sequence. Proof Having chosen ¢ > 0, we can chose n, > m so that, for every n >mn,, —¢<, and x” < 4 &. For these #’s we then have, ipso facto, —e <®, < + ¢, that is |x, | 0 has been chosen arbitrarily, then by hypothesis (cf. 10, 12) a number n, and a number #, exist such that for every "> n,, |, | < 5 ,and forevery n > n,, jo) <5 If n,is a number =n, and > u,, then for 2 > 2, 90] = yy | | [2] < ff =e (y,) is therefore a null sequence. Since, by 26, 3~for 10, 5), (— x’) is a null sequence if (z,’) is, (y,) = (x, — =,’) is then by the above also a null sequence, i. e. we have the theorem: yu! ©Theorem 2. If (x,) and (x) are null sequences, then so is (®)) = (x, — =). Or briefly: null sequences “may” be subtracted term by term. Remarks. 1. Since we may add fwo null sequences term by term, we may also do so with three or any definite number of null sequences. For supposing this prov- ed for (p—1) null sequences (2), (@”), -++, (z?7V), i. e. supposing the sequence (@a +2" + +e +2271) to be already recognised as a null sequence, Theorem 1 ensures that the sequence (z,), for which Tp = (2 + see +421 + =P, is also a null sequence. The theorem thus holds for every fixed number of null sequences. 2. That two null sequences “may” also be multiplied term by term, is immediately clear from 26, 1, since null sequences, by 10, 11, are necessarily bounded. 8. Term by term division, on the coatrary, is in general not allowed, as is already obvious, for instance, from the fact that when z, = 0, SH is con- n 1 1 oii stantly =1. If we take z, =o rd dn then the ratios oi do not even pro- n vide a bounded sequence. s By 3,11, 4. § 7. Powers, roots and logarithms. Special null sequences. 47 4. In the case of other sequences (z,) also, little can be said in the first instance about the sequence = of the reciprocal values. The following is n an obvious, but often useful theorem: OTheorem 3. If the sequence (|x, |) of absolute values of the terms of (wy) have a positive lower bound, — if, therefore, a number py > 0 exists, such that for every m, Tn = ¥ = 0, then the sequence (&) of reciprocal values is bounded. n In fact, from |a, | =y > 0 it at once follows that for ral we have 1 SK Zn a for every mu. In order to increase the scope both of the application of our con- cepts and of the construction and solution of examples, we insert a paragraph on powers, roots, logarithms and circular functions. § 7. Powers, roots and logarithms. Special null sequences. As, in the discussion of the system of real numbers, it was not our intention to give an exhaustive treatment of all details, but rather to put fundamental ideas alone in a clear light, assuming as known, thereafter, the body of arithmetical rules and concepts, with which after all everyone is thoroughly conversant, so here, in the discussion of powers, roots and logarithms, we will restrict ourselves to an exact elucidation of the definitions, and then assume known the details of their application. I. Powers with integral exponents. If is an arbitrary number, we know that the symbol z* for positive integral exponents k is defined as the product of k factors, all equal to #. Here we have therefore nothing substantially new, but only an abbreviated notation for something we know already, If x50, it is convenient to agree, besides, that 1 2° represents the number 1, 2% the number or (h=1,2,3,. oh so that x? is defined for every integral $30. For these powers* with integral exponents, we merely emphasize the following facts: 1. For arbitrary integral exponents p and g¢ (20) the . three fundamental rules hold: LP +BY == PEL; z?.y? = (x y)?; (Zs nr, * 2P is a power of base x and exponent p. This continental use of the word power cannot be here dispensed with, in spite of the slight ambiguity resulting from by far the most frequent use of the word in English to designate the exponent. This sense should be entirely discarded from the reader’s mind, notably for § 35,22 and others. (Tr.) 29. 48 Chapter II. Sequences of real numbers. from which all further rules may be deduced, which regulate calcu- lations with powers?®. 2. Since, in a power with integral exponent, merely a repeated multiplication or division is involved, its calculation has of course te be effected by 18 and 19. If therefore x is positive and defined for instance by the nest (x, |y,), with all its endpoints > 0 (cf. 15, 5), then we have simultaneously with pe (2, | 3.) = (xE | 4%) at once, for all positive integral exponents; and similarly — with appropriate restrictions — for # <0 or 2 Lo. 3. For a positive x we have furthermore > rS1 as we at once deduce from 251, if we multiply (v. 3,1, 3) by BY oi wig af according as And quite as simply we find: If x, x, and the integral exponent k are positive, then BS at according as = 375 4. For positive integral exponents n and arbitrary a and b we have the formula (a+ b)"=a"+ (3) an=1p 4 (5) ar—2p% I-... re where 3 , for 1 on We write Isai and call & the k®™ root of a. Proof. One such number may immediately be determined by a nest of intervals, and its existence thereby established: We use the decimal-section method. Since 0 = 0 < a, but, p denoting any positive integer > a, p* > p > a, — there is one and only one integer g > 0 for which 10 Fn lib1Y, The interval J, determined by g and (g 4 1) we divide into 10 equal parts and obtain, in the manner now repeatedly worked out, a defi nite one of the digits 0, 1, 2, ..., 9, — which we may denote, say, by z,, — and for which lo 50) 0, it at once follows by 29, 2 that § = (of | 42) But, by construction, 2¥ 0 and a, > 0, then Va Va, , according as a,. — Further we have the VIIA a VINA n— Theorem 2. If a > 0, then va) is a monotone sequence; and we have, more precisely, iy 3 arvas Vaz i21,; if a >1, but ol, 3. da 1 involves a?! >a" >1, and therefore by the preceding theorem, taking (xn - 1)" roots, Va >"Va>1. Since for a <1 all the inequality signs are reversed, this proves the whole statement. — Hence finally we deduce the Theorem 3. If a > 0, then the numbers x, = Va —1 form a null sequence (monotone by the preceding theorem). Proof. For a =1, the assertion is trivial, as then x, =0. H nn. a >1, and therefore Va >1, 1.6 2 =Va — 1 > 0, then we reason i" — as follows: By the inequality of Bernoulli (v.10,7), Va =1-4z, gives a=QA+4z)>1tnz,>nz,. Consequently z, = |z, | < =, therefore (z,), by 26,1 or 2, is a null sequence. If 0<.qa <1, then 3 >1, and so, by the result obtained, Fo is a null sequence. If we multiply this term by term by the factors Va, ”n— — which certainly form a bounded sequence, as a < Va <1, — then it at once follows, by 26, 1, that (1 I) and therefore also (z,), : is a null sequence, — g.e. d. § 7. Powers, roots and logarithms. Special null sequences. 51 III. Powers with rational exponents. We again regard as substantially known, in what manner one may pass from roots with integral exponents to powers with any rational 2 exponent: By a?, with integral $50, q > 0, we mean, for any posi- tive a, the positive number uniquely defined by » » a2), » If p> 0, then ¢ may also be = 0; a? must then be taken to have the value 0. With these definitions, the three fundamental rules 29, 1, i. e. the formulae : ad = ar; = (a by; (y = gr? remain unaltered, for any rational exponents, and therefore calculations with these powers are formally the same as when the exponents are integers. These formulae contain, at the same time, all the rules for working with roots, since every root may now be written as a power with a rational exponent. — Of the less known results we may prove, as they are particularly important for the sequel, these theorems: Theorem 1. When a > 1, — then a" > 1, if, and only if, r > 0. 32. Similarly, when a <1 (but positive), then a" is <1 if, and only if, r>0. Proof. By 31,2, a and Vo are either both greater or both less . 7 \p than 1; by 29 the same is true of # and (Va) == if ‘and only if 2>0, Theorem 1a. If the rational number » > 0, and both bases are positive, then ra, according as esa, The proof is at once obtained from 31,1 and 29, 3. Theorem 2. If a > 0, and the rational number » lies between the rational numbers ¥ and 7”, then a’ also always lies between a” and a”1%, and conversely, — whether a be <, =or>1, and ¥ <, = >| Proof If firstly, a >1 and 7 <7 <7", then ’3 a’ ’ ol @ Smigl. gf? = op, a 13 The term “between” may be taken, as we please, either to include or exclude equality on both sides, — excepting when a =1, and therefore all the powers a” also =1. 33. 59 Chapter II. Sequences of real numbers. By the preceding theorem, this already proves the validity of our state- ment for this case, and in the other possible cases the proof is quite as easy. — From this proof we deduce, indeed, more precisely, the- Theorem 2a. If a >1, then fo the larger (rational) exponent also corresponds the larger value of the power. If a <1 (but positive) then the larger exponent gives the smaller power. — In particular: If the (positive) base a ==1, then different exponents give different powers. — Hence we deduce, further, Theorem 3. If (r,) is any (rational) null sequence, then the numbers ; 2, =a" =1, (a> 0) also form a null sequence. If (r,) is monotone, then so is (x). — ni1 Proof. By 31,3, {a a 1) and (VE = 1) are null sequences. If therefore & > 0 be given, we can so choose n, and #n, that In for mm, Va it] zo. and for zn > n,, If m is an integer larger than both #, and #,, then the numbers 1 1 (5 —_ 1) and (They) both lie between — & and J ¢, 1. e. 1 1 am and ¢ ™ lie between 1 — ¢ and 1 Je. By Theorem 2, a” then lies between the same bounds, if » lies be- tween — > and + hs By hypothesis we can, however, so choose #,, that for every un > n,, 1 1 i pic 8 =n, wo for n> n,, a™ is therefore between 1 —¢ and 1-4-¢. Hence, for these #'s, | an —1 22.85 proving that (¢" —1) is a null sequence. — That it is monotone, if (r,) is, follows immediately from Theorem 2a. These theorems form the basis for the definition of IV. Powers with arbitrary real exponents. For this we first state the Theorem. If (x, |y,) is any nest of intervals (with rational end- points) and a is positive, then for a>1, o=(a"a’") and for a <1, o= (a |a"™) §7. Powers, roots and logarithms. Special null sequences. 58 is also a nest of intervals. And if (x,|y,) is rational valued and =r, ther 6 = 4. Proof. That in either case the left endpoints form a monotone ascending sequence, the right endpoints a monotone descending se- quence, follows at once from 32, 2a. By the same theorem, a2 < in in the one case (a >1) and a’# < 4 in the other (a <1), for every x. Finally, that in both cases the lengths of the intervals form a null sequence, follows, with the aid of 26, from lan — a" — | an" — 1] .a™; for here the first factor, by 32, 3, is a null sequence, because (y, — x) is by hypothesis a null sequence with rational terms; and the second factor is bounded, because for every = Oca? L in the one case (a > 1), =u in the other (a <1). a Now if (x,|y,) =, then 7 lies between z, and y,, for every z, and so by 82, 2, a’ lies between ¢"» and a¥", for every un; hence by. § 5, Theorem 4, necessarily oc = a’. In consequence of this theorem, we may agree to the following Definition. If a >0, and go=(x |y,) is an arbitrary real number, then: : = (|) # a1 a’ =o, i e. pile if a<14, This definition can of course only be regarded as appropriate, if the concept of a general power thereby determined obeys substan. tially the same laws as the type of power so far considered, that with rational exponents. That this is so, in the fullest sense, is shewn by the following considerations. 1. For rational exponents, the new definition gives the same result $4. as the old. 2.1 g==g, then a2 = a2’®, 14 This combination 38 of theorem and definition is, from the point of view of method, of exactly the same kind as those set forth in 14—19: What is demonstrable in the case of rational exponents is raised, in the case of arbitrary exponents, to the rank of a definition, — whose appropriate- ness has then to be verified. 15 This assertion, formally rather trivial in appearance, when put some- what more explicitly, runs thus: If (z,|y,) =¢ and (x, |y,’) =o’ are two nests of intervals, which may be regarded as equal in the sense of 14, then so are those nests of intervals equal (again in the sense of 14), which by Definition 33 give the powers a2 and a?. A | 3% 54 Chapter Il. Sequences of real numbers. 3. For two arbitrary real numbers gp and ¢’, and positive a and b, the three fundamental rules at.af s=qeie; (a0-U0)s=(ab)e; (a2) == aot, hold, so that with the general powers now introduced we may cal- culate formally in precisely the same way as with the special types- hitherto used. Into the extremely simple proofs of these facts we will, as emphasized on p. 17, not enter further®; we will also, so far as concerns the extension of theorems 82, 1—3 to general powers, now . immediately possible, content ourselves with the statement and a few 35. indications of the proof. We have therefore the theorems, generalized from 32, 1—3: Theorem 1. When a > 1, we have a® > 1 if, and only if, 9 > O. Similarly, when a <1, (but positive), we have a? <1 if, and only if, o> 0. For by 82,1, we have e. gz. for a > 1, a® > 1 if, and only if z,>0. Theorem la. If the real number @ is > 0, and both bases are aes ae positive, then a°s af, according as a 2 ay. Proof by 32, 1a and 15. Theorem 2. If a > 0 and o is between ¢' and 0”, then a® is al- ways between a? and a?’. — The proof is precisely the same as 32, 2. It yields, more exactly, the Theorem 2a. If a> 1, then to the larger exponent corresponds the larger value of the power; if a <1 (but positive), then the larger exponent gives the smaller power. In particular: If a <= 1, then different exponents give different powers. — And from this theorem, exactly as in 82, 3, follows the final 18 As a model we may sketch the proof of the first of the three fundamental rules: If o= (nv; |, and o'= = (@' | ¥u') then by 16, 0 +0" = (x, +2," | ¥» + 3.) and therefore — we assume a=1 — > a? = (a a¥), 2% (a a¥n’), athe = (ot | a¥ntin’), Since all endpoints (as powers with rational exponents) are positive, we have, by 18, a2.a® 2 (a®n.a® | a¥n.q¥"). Since, however, for rational exponents, the first of the three fundamental rules has already been seen to hold, this last nest of intervals is not only equal, in the sense of 14, to that define a?t? | but even coincides with it term by term. §7 Powers, roots and logarithms. Special null sequences. D5: Theorem 3. If (o,) is any null sequence, then the numbers x, =a’ —1 5 (a> 0) form a null sequence. If (o,) is monotone, then so is (,). As a special application, we may mention the Theorem 4. If (x,) is a null sequence with all its terms positive, then for every positive e, aw z, n ? 18 also the term of a null sequence. — Thus £2 for evety a >0 is a null sequence. ” 1 Proof. If ¢ > 0 be given arbitrarily, e® is also a positive number. By hypothesis, we can choose n, so that, for every un > n,, 1 2, | ==, ny, by 3, 1a, we then also have, however, Bt = nts which at once proves the whole statement. The above theorems comprise the main principles used in cal- culations with generalized powers. V. Logarithms. The foundation for the definition of logarithms lies in the Theorem. If a> 0 and b> 1 are two real, and in all further 36. respects quite arbitrary numbers, then one and only one real number & always exists, for which bi==p. Proof. That at most one such number can exist, already follows from 35, 2a, because the base b with different exponents cannot give the same value g. That such a number does exist, we show con- structively, by assigning a nest of intervals which determines ‘it, — thus for instance by the method of decimal sections: Since b > 1, Oo" — () is a null sequence, by 10, 7, and there exists, conse- quently, since a and x are positive, natural numbers p and ¢ for which Wn. ond. ldo Mima, If, now, we consider the various integers between — p and gq in succession, as exponents of bp, there must be one, and can be only one — call it g¢ — for which Wa, Bit ily, 56 Chapter II. Sequences of real numbers. The interval Jy =g...(g-} 1) thereby determined we divide into 10 equal parts and obtain, just as on p. 49, a “digit” z,, for which 21 +1 g+— s+ 0.5 bn YT, b By repetition of the process of subdivision we find a perfectly definite nest of intervals = z Zi : Z,=g+ 5+... + 4 E = (x, | Yu) with z Zz Zp +1 | 1 n n Yn=&T 1g T ++ T fori T pon 2 for which pn < 4.< py» for every m, — for which, therefore, in accordance with 33, = This theorem justifies us in the following Definition. If a > 0 and b > 1 are arbitrarily given, then the real number &, uniquely determined by bé=a ts called the logarithm of a to the base b; and, symbolically, S=loz, a. (g ts also called the characteristic, and the set of the digits z,, Zys 2g eee the mantissa, of the logarithm.) We speak of a system of logarithms, when the base b is assum- ed fixed once for all and the logarithms of all possible numbers are taken to this base b. The suffix b in log, is then usually omitted as superfluous. Very soon a particular real number, usually denoted by e, appears quite naturally as the most convenient for all theo- retical considerations; the system of logarithms built up on this base is usually called the system of natural logarithms. For practical purposes, however, the base 10 is, as we know, the most convenient; logarithms to this base are called common or Briggs’ logarithms. These are the logarithms found in all the ordinary tables!”. The rules for working with logarithms we assume, as we did with powers, to be already known, and content ourselves with a mere mention of the most important of them. If the base b > 1 is arbitrary, | J 17 As a matter of course, a system of logarithms may also be built up on a positive base less than 1. This, however, is not usual. The first loga- rithms calculated by Napier in 1614 were, however, built up on a base b<1, which presents some small advantages, particularly for logarithms of trigono- metrical functions. § 7. Powers, roots a d logarithms. Special null sequences. 57 ‘but assumed fixed in what follows, and if a, a’, a”... denote any positive numbers, then 1. log (a’ a’) = log a’ -\- log a”. 37. 12. log 1=0; log + — —loga; logh=1. 3. loga¢ =gloga (go arbitrary, real). 4. loga Slog a’, according as a 5 a’; in particular, = i > 5. loga Z 0, according as a 21. 6. If b and b, are two different bases (> 1), and & and ¢&, the logarithms of the same number a to these two bases, i. e. f==log,a, §& =log,a, then ? ? : 2 == 50102..0,, a as follows at once from (a =)b% = b,%, by taking logarithms on both sides to the base b and taking account of 87, 2 and 3. 1 . 1 7 (55g) n=12.3,4.. is a null sequence. In fact 225 <= 2 1 provided log n > > that is, n> 0". VI. Circular functions. To introduce the so-called circular functions (the sine of a given angle’, with the cosine, tangent, cotangent etc.) in an equally strict manner, i. e. avoiding on principle all reference to geometrical in- tuition as element of proof and founding solely on the concept ot the real number, is at this stage not yet possible. This question will be resumed later (§ 24). In spite of this, we will unhesitatingly enlist them to enrich our applications and enliven our examples (but of course never to prove general propositions), in so far as their Tow ledge may be presupposed from elementary work. Thus e. g. the following two simple facts can at once be ascertained: 37a. 1. If a, tg, ..., &, ... are any angles (that is to say, any numbers), then (sin ez) and (cos «,,) are bounded sequences; and 18 Angles will in general be measured in radians. If in a circle of radius unity we imagine the radius to turn from a definite initial position, then we measure the angle of turning by the length of the path which the extremity of the moving radius has traversed — taking it as positive when the sense of turning is counterclockwise, otherwise as negative. An angle is accordingly a pure number; a straight angle has the measure 4m or —a, a right angle the measure pi or - To every definitely placed angle there belongs an infinite number of measures which, however, differ from one another only by integral multiples of 2a, i. e. by whole turns. The measure 1 belongs to the angle, the arc corresponding to which is equal to the radius, and which there- fere in degrees is 579 17’ 44-8 nearly. 38. 58 Chapter II. Sequences of real numbers. {= Zr) ( 2) —=*) and —r n n are (by 26) null sequences, for their terms are derived from those of the 2. the sequences null sequence 3) by multiplication by bounded factors. VII. Special null sequences. As a further application of the concepts now defined, we will examine a number of special sequences: C1. If |a| <1, then besides (a™) even (na™) is a null sequence. Proof. Our reasoning is analogous to that of 10, 71%: For a=0, the assertion is trivial; for 0 <|a| <1, we may write, with go > 0, 1 1407 Yo 4 {2 Jorl \n jal= ro and therefore |a”]|= Since here in the denominator each term of the sum is positive, we have for every # > 1, : y therefore: |sia®|< allo ( 2) o m—1)p 2 2 n - |na™| 1-1- or The result thus proved is very remarkable: it asserts, in fact, that for a large n the fraction is very small, and its denominator n T+o" therefore very much greater than its numerator. This denominator is however constant (=1) for 9 =0, and when g is very small (and positive), it only increases very slowly with n. Nevertheless, our result shows that provided only n be taken sufficiently large, the deno- minator is very much larger than the numerator. The point #,, from lies below a given ¢ — we found n, = 1} ‘hi 2 —_— Meas — which na” | EL I does indeed lie very far to the right, not only when &, but also when 0—= th — 1, is very small (i. e. |a| very near to 1). Substantially this 19 Except that @ and p need no longer be rational. i YR § 7. Powers, roots and logarithms. Special null sequences. 59 and only this is true: However |a|< 1 and ¢> 0 may be given, we have always, from a readily assignable point onwards, |na”"| < &2 From this result many others may be deduced, e. g. the still more paradoxical fact: ©2. If |a| <1 and « real and arbitrary, then (n*a™) is also a null sequence. Proof If #0, then this is evident from 10, 7, because of 1 26.2; if «> 0, write jale =a, so that by 39, 1a, the positive number a, is also <1. By the preceding result, (ng,”) is a null sequence. By 39, 4 [a7 Le wlan or’ Iucqv| thercfore, finally, (by 10, 5), n*a™ itself is also the term of a null sequence. (If we think of « as given very large and |a| only very little less than 1, this result is of course still more remarkable than the preceding.) 3H 6>0, then ae n ) is a null sequence, to whatever base b> 1 the logarithms are taken. Proof. Since b> 1, 6 > 0, we have (by 33, 1a), b° > 1. There: fore Li) is a null sequence, by 1. Given ¢ > 0, we have conse- quently from a certain point onwards, — say for every n >m — Sh b"™, logn, and a fortiori g + 1,is > m. Hence the last value above, with our choice of m, is But, in any case, £ : log »n Kolnpry, Le zg « 2 lor every a> 5, =D", 20 Writing as above (dees, |ne®|= , we may also say: 1+ Tre (1+ po) becomes — for a positive po — more pronouncedly large, or, also more pronouncedly infinite, than = itself, — by which we again (cf. '?, 3) mean nothing more and nothing less than that our sequence is precisely a null sequence. — For future reference we remark here that the results proved in 1 and 2 are also valid for a complex a, provided only |a| <1. 21 With the same change of notation as above, we may say here: (1-}-o)" becomes more pronouncedly infinite than every (fixed) power however large of » itself. 22 Ulog nm becomes less pronouncedly lavge than every power, however small, (but determinate and positive), of » itself.” - 60 Chapter II. Sequences of real numbers. 4. If o and B are arbitrary positive numbers, then (42) is a null sequence —, however large « and however small f may be. Proof. By 3, tw A >0; by ble Ir 395, 4, therefore, so is the given sequence. ) is a null sequence, because Bo 5. Cystine) is a null sequence. (This result is also very remarkable. For when # is large, we have a large number under the Vv; the exponent of the Y is, it is true, also large; but it is not at all evident @ priors which of the two — radicand or exponent — will, so to speak, prove the stronger.) "n .- Proof, For n>1, we certainly have Van >1, therefore Nn... x, = Vu —1 cetiainly > 0. Hence in n=Q+z)=1+ (7), +... +(2) an all the terms of the sum are positive. Consequently we have, in particular, n a n(p=D 2 n> {zn nS or 2 2 4 2 ee 2 Zn = wo eT Hence 2 |=, | <=’ n2 n--- so that (zx) = Van — 1 is in fact by 26, 3 and 35, 4 a null sequence. 6. If (x,) ts a null sequence whose terms are all > — 1, then for every (fixed) integer k, the numbers k x, =Y1 +x, —1 also form a null sequence. 23 «Every power of log #, however large, (but fixed) becomes less pronouncedly large than every power of x itself, however small (but fixed). n : 5 for (n — 1) which it cannot exceed, is an artifice often useful in simplifying calculations. 25 By the assumption that all x,’s > —1, we merely wish to ensure that the numbers x,’ are defined for every mn. From a definite point onwards this is automatically the case, since (x,) is assumed to be a null sequence and therefore from some point certainly |x, |< 1, and hence z, > — 1. 24 The substitution, when # > 1, of the value n — §7. Powers, roots and logarithms. Special null sequences. 61 Proof. From the formulae set forth on p. 21, Footnote 11, where i we put a = V1-+ x, and b=1, it follows that ?® gC Ln Zz, == Er Slr therefore, since the terms in the denominator are all positive and the last is 1, EMESENE whence, by 26, the statement at once follows. 7. If (x,) is a null sequence of the same kind as in 6., then the numbers : Y= log (1 + 25) also form a null sequence. Proof. If b> 1 is the base to which the logarithms are taken, and ¢ > 0 is given, we write bp —i=2, 1-0" =g¢, and denote the smaller of the two (necessarily positive) numbers eg, ‘and ¢, by ¢. We then choose n, so large, that for every # > n,, |#,| < ¢&. For those #'s we have, a fortiori, — tL Cty hoe UT <1 LU, bE therefore (by 35, 2 or 37, 4) |y, | =|log(1 +2.) | 2, since for k =1 the assertion is trivial. Eo 39. 62 Chapter II. Sequences of real numbers. § 8. Convergent sequences. Definitions. So far, when considering the behaviour of a given sequence, we have been chiefly concerned to discover whether it was a null sequence or not. By extending this point of view somewhat, in a manner which readily suggests itself, we reach the most important concept of all with which we shall have to deal, namely that of the convergence of a sequence. We have already (cf. 10, 10) described the property which a se- quence (z,) may have, of being a null sequence, by saying that its members become small, become arbitrarily small, with increasing n. We may also say: Its terms, as # increases, approach the value 0, — without, in general, ever reaching it, it is true; but they approach arbitrarily near to this value in the sense that the values of its terms (that is to say, their differences from 0) sink below every number &(> 0), how- ever small. If we substitute for the value O in this conception any other real number &, we shall be concerned with a sequence (x) for which the differences of the various terms from the definite number & — that is to say, by 3,1, 6, the values |x, — &|, — sink, with increas- ing n, below every number & > 0, however small We state the matter more precisely in the following © Definition. If (x,) is a given sequence, and if it is related to a definite number & in such a way that (=, = £) : forms a mull sequence, then we say that the sequence (x,) converges lo &, or that it is convergent with the limiting value &, or that its terms approach the (limiting) value &, tend to &, have the limit &. And this fact is expressed by the symbols Wa—>8 or limo,=2§. To make it plainer that the approach to & is effected by taking the index n larger and larger, we also frequently write Xpn—8 for n—oo or limax,=§> n->o Including the definition of a null sequence in the new definition, we may also say: x, —& for n—oco (or limzx, = §&) if for every chosen e > 0, we nro : can always assign a number ny, =n, (e), so that for every n > n,, we have jo, — él 0 Va— 1 and Vi 1. 4. If (z,|y,) =o, then 2,— 0 and y,—>co. For both |#,—o| and also |y,—o]| are <|y,—a,], so that both, by 26, 2, form null sequences together with (y, — x,). 5. For x, =1 cial that is for the sequence 2 Eh 8 3 20). FL BL Gon 2, —1, for |2,—1| — forms a null sequence. 6. In geometrical language, x, — & means that all terms with sufficiently large indices lie in the neighbourhood of the fixed point £&. Or more precisely (cf. 10, 13), in every e-neighbourhood of &, the whole of the terms, with at most a finite number of exceptions, are to be found. — In applying the mode of re- presentation of 7, 6, we draw parallels to the axis of abscissae, through the two points (0, &§ 4- ¢) and may say: z, — &, if the whole graph of the sequence (z,), with the exception of a finite initial portion, lies in every e-strip (however narrow). 7. The lax mode of expression: “for n=00, ,=§" instead of z,—§, should be most emphatically rejected. — For an integer n = 00 does mot exist and x, need never be =&. We are concerned merely with a process of approxi- mation, sufficiently clear from all that precedes, which there is no ground whatever for imagining completed in any form. (In older text books and writ- ings we frequently find, however, the symbolical mode of writing: “lim x, = &”, n=w - to which, since it is after all meant only symbolically, no objection can be taken, — excepting that it is clumsy, and that writing “n = 00” must neces- sarily create some confusion regarding the concept of the infinite in mathematics. 8. If x, —&, then the isolated terms of the sequence (z,) are also called approximations to &, and the difference & — x, is called the error corresponding to the approximation z,. 9. The name “convergent” appears to have been first used by J. Gregory (Vera civculi et hyperbolae quadvatura, Padua 1667), and ‘‘divergent” (40) by Bernoulli (Letter to Leibnitz of 7. 4. 1713). To the definition of convergence we at once append that of divergence: © Definition 1. Euvery sequence which is not convergent in the sense 19, = of 39, is called divergent. 3 Frequently this is expressed more briefly: In every e-neighbourhood of & “almost all” terms of the sequence are situated. The expression “almost all” has, however, other meanings, e. g. in the Theory of Sets of Points. 64 ! Chapter II. Sequences of real numbers. With this definition, the sequences 6, 2, 4, 7, 8, 11 are certainly divergent. Among divergent sequences, one type is distinguished by its particularly simple and transparent behaviour, e. g. the sequences (n?), (n), (a") for a > 1, (logn), and others. Their common property is evidently that the terms increase with increasing » beyond every bound, however high. For this reason, we may also say that they tend to | co, ~ or that they (or their terms) become infinitely large. This we put more precisely in the following Definition 2. If the sequence (x,) has the property that, given an arbitrary (large) positive number G, another number ny can always be assigned such that for every n > mn, 2, >G % — then we shall say that (x,) diverges to -- oo, tends to + oo, or is definitely divergent with the limit + 00%; and we then write x, — +o (for n—o0) or limx,= 4oc0 or limz, = + oo. n—>x We are merely interchanging right and left by defining further: Definition 3. If the sequence (x,) has the property that, given an arbitrary negative number — G (large in absolute value), another number n, can always be assigned such that for every n> mn, z,<—0C, then we shall say that (x,) diverges to — oo, tends to — oo or is definitely divergent with the limit — oO, and we write x,— — © (for n—00) or limz,= —o00 or Lig 0 xo Remarks and Examples. 1. The sequences (rn), (n?), (n*) for «> 0, (logn), (log n)* for 0 tend to + oc; those whose terms have these values with the negative sign tend to — 00. : 9.1In general: If x, — + 00, then z,’ =— 2, —> — 00, and conversely. — It is therefore sufficient, substantially, to consider divergence to + QO in what follows. 3. In geometrical language, z, — + 0O means, of course, that however a point G (very far to the right) may be chosen, all points z,, except at most a finite number of them, remain beyond it on the right. — With the mode of 4 Notice that here not merely the absolute values |x, |, but the numbers wz, themselves, are required to be > G. 5 It is sometimes even said, — with apparent distortion of facts, — that the sequence converges to +00. The reason for this is that the behaviour described in Definition 2 resembles in many respects that of convergence (39). We will not, however, subscribe to this mode of expression, although a mis- understanding would never have to be feared. — Similarly for — oo. § 8. Convergent sequences. 65 representation in '7, 6, it means that: however far above the axis of abscissae- we may have drawn the parallel to it, the whole graph of the sequence (z,) — excepting a finite initial portion, lies still further above it. 4. The divergence to -- 00 need not be monotone; thus for instance the sequence 1, 2%, 2,228,623, 4, 24,.,.,,%,2%,... also diverges to 00. 5. The succession 1, —2, +3, —4, ..., (—=1)""1#n,... does not diverge to +00 or to —00. — This leads us to the further Definition 4. A sequence (x), which either converges in the sense of definition 39, or diverges definitely in the sense of the defini- tions 40, 2 and 3, will be said to behave definitely (for n— oo). All other sequences, which therefore neither converge, nor diverge defini- tely, will be called indefinitely divergent or, for short, indefinite®. Remarks and Examples. 1. The sequences [(— 1)*], [(—2)"], (a®) for a < —1, and likewise the se- quences 0,1,0,2,0,8,0,4,... and 0, —1, 0, —2, 0, — 3, ..., as also the se- quences 6, 4, 8 are obviously indefinitely divergent. 2. On the contrary, the sequence (|a”|) for arbitrary a, and, in spite of all irregularities in detail, the sequences (37+ (—2)"), (n+ (—1)"log un), (n® 4 (— 1)" m), show definite behaviour. 3. The geometrical interpretation of indefinite behaviour follows imme- diately from the fact that there is neither convergence (v. 89, 6) nor definite divergence (v. 40, 3, rem. 3). 4. Both from z, —-+ 00 and from z,— — 00 it follows, provided every 1 term = 07, that ==> 0} for lus G=n evidently implies 1: < é&. — On n n the other hand, x, —> 0 in no way involves definite behaviour of (=) . Zn — 1)» Example: For at z , we have z,— 0, but =) indefinitely diver- n gent. — We have however, as is easily proved, the Theorem: If (x,) is a null sequence whose tevms all have the same sign, then the sequence &) is definitely divergent; — and of course to + oO or n — 00, according as the x,’s ave all positive or all negative. 8 We have therefore to consider three typical modes of behaviour of a sequence, namely: a) Convergence to a number &, in accordance with 39; b) divergence to 4 00, in accordance with 40, 2 and 3; c) neither of the two. — Since the behaviour b) shows some analogy with a) and some with c), modes of expressions in use for it vary. Usually, it is true, b) is reckoned as divergence (the mode of expression mentioned in the last footnote cannot be consistently maintained) but “limiting values” +4 00 and — oO are at the same time spoken of. — We therefore speak, in the cases a) and b), of a de- finite, in the case c) of an indefinite, behaviour; in case a), and only in this case, we speak of convergence, in the cases b) and c) of divergence. — Instead of “definitely and indefinitely divergent”, the words “properly and im- properly divergent” are also used. Since, however, as remarked, definite di- vergence still shows many analogies to convergence and a limit is still spoken of in this case, it does not seem advisable to designate this case precisely as that of proper divergence. ” From some place onwards this is certainly the case. 41. —- 66 Chapter 1I. Sequences of real numbers. To facilitate the understanding of certain cases which frequently occur, we finally introduce the following further mode of expression: O Definition 5. If two sequences (x,) and (y,), not necessarily con- vergent, are so related to ome another that the quotient Zn < Yn tends, for n— -}-00, to a definite finite limit different from 2er08, then we shall say that the two sequences are asymptotically proportional and write briefly Tn Yn: If in particular this limit is 1, then we say that the two sequences are asymptotically equal and write, more expressively : rT, 2, Thus for instance A 1 Vi+l~n, log(®n®+23)~logn, Vutl—yn ~~ n 1+24-cc4n~on?, 124-220... Ln 1 m3, These designations are due substantially to P. du Bois-Reymond (Annali di matematica pura ed appl. (2) IV, S. 338, 1870/71). To these definitions we now attach a series of simple, but quite fundamental Theorems on convergent sequences. © Theorem 1. A convergent sequence determines its limit quite uniquely”. ; ; Proof. If z —&, and simultaneously x, — &, then (z, — &) and (x, — &) are null sequences. By 28, 2, (= 8 ~~ = —8 is then also a null sequence, i.e. §=¢, q.e. d.1° 8 x, and y, must then necessarily be == 0 from some place onwards. This is not required for every » in the above definition. 9 A convergent sequence therefore defines (determines, gives...) its limit quite as uniquely as any nest of intervals or Dedekind section defines the number to which it corresponds. Thus from this point we may consider a real number as given if we know a sequence converging to it. And as formerly we said for brevity that a nest of intervals (x, |¥,) or a Dedekind section (4|B) or a radix fraction Zs a real number, so we may now with equal right say that a sequence (w,) converging to & ¢s the real number §&, or symbolically: (z)=26. For further details of this conception, which was used by G. Cantor to construct his theory of real numbers, see pp. 77 and 93. 10 The last step in our reasoning, by which the reader may at first sight be taken aback, amounts simply to this: If with respect to a definite numerical value ¢ we know that, for every &¢>0, we always have |a| 0, assign a number wm, such that for every n > m E—eK, then |£| — K>0 and therefore from some place onwards in the sequence, |8] ~ lull, Elis XK and therefore to, > K, which is contrary to the meaning of K. OTheorem 2a. x, —& implies |x, |— | &]. Proof. We have (v. 3,11, 4) Jol =12l] glo, ~ ls therefore (|#,| — |£|) is by 26, 2, a null sequence when (2, — £) is. © Theorem 3. If a convergent sequence (x,) has all its terms diffe- vent from zero, and if its limit & is also == 0, then the sequence £3) n ts bounded; or im other words, a mumber y > 0 exists, such that x, | =y > 0 for every m; the numbers |x, | possess a positive lower bound. Proof. By hypothesis, |&| = &>0, and there exists an integer m, such that for every n > m, |x, — £| < ¢ and therefore |x, | >1|&|1 If the smallest of the (m -}- 1) positive numbers |z, |, |=,|, ..., |=, i L|&| be denoted by y, then y > 0, and for every un, |z,|=> 7, il CS Kss 5 q.e: 4d. If, given a sequence (z,) converging to &, we apply to the null sequence (x, — &) the theorems 27,1 to 5, then we immediately ob- tain the theorems: , necessarily have ¢ =0. For if a 3 0, then |«|>0. Choosing for & the num- ber }| «|, we should certainly not have |e |< e. Similarly, if we know of a definite numerical value o that, for every ¢>0, we always have ao < K-&, then we must have further « << K. The method of reasoning involved here: “If for every > 0, we always have |e | < &, then necessarily « = 0” is precisely the same as was constantly applied by the Greek mathematicians (cf. Euclid, Elements X) and later called the method of exhaustions. , 11 Here the sign of equality in “|£|< K” must not be omited, even when, for every Wy Bn) IL, 12 For % rh. all the z,’s are therefore necessarily 0. 68 Chapter II. Sequences of real numbers. OTheorem 4. If (x) is a subsequence of (x), then 2,—& implies w'—E. © Theorem 5. If the sequence (x,) can be divided into two sub- sequences'® of which each converges to &, then (x) itself converges to &. OTheorem 6. If (x) is an arbitrary rearrangement of x,, then x, —& implies wx —E. OTheorem 7. If x,—¢& and (x) results from (x,) by a finite number of alterations, then x, —&. > OTheorem 8. If x,/—¢& and x,” —&, and if the sequence (x) is so related to the sequences (x) and (x.”) that from some place onwards, (i. e. for every m =m, say,) then x, — E. Calculations with convergent sequences are based on the following four theorems: : OTheorem 9. x, —& and y,— 0 always implies (x,+v,)— E+, and the corresponding statement holds for term by term addition of any fixed number — say p — of convergent sequences. Proof. If (xz, —£) and (y, — 7) are null sequences, then so, by 28,1, is ((@, +, — (+19). In the same way, 28, 2 gives the OTheorem 9a. z,—& and y,— 1, always implies (x, — y,)—&—1). OTheorem 10. x,—¢& and y,— 7, always implies x,y,—&), and the corresponding statement holds for term by term multiplication of any fixed number — say p -— of convergent sequences. In particular: x, —§& implies cx,—c&, whatever number » denote. Proof. We have By, —=Eg=ly —8y L(y —Ne; and since here on the right hand side two null sequences are multi- plied term by term by bounded factors and then added, the whole expression is itself the term of a null sequence, g. e. d. OTheorem 11. z,— & and y,— 17 always implies, if every x, == 0 and also £0, LL o£ Proof. We have Yo No Yuku] Onn E-(m—8)y po § Zn XE 13 Or three, or any definite number. § 8. Convergent sequences. 69 Here the numerator, for the same reasons as above, represents a null sequence, and the factors a are, by theorem 3, bounded. Therefore the whole expression is again the term of a null sequence. — Only a particular case of this is the °Theorem lla. x,—¢& always implies, if every x, and also & are == 0, 1 lag Zn These fundamental theorems 8—11 lead, by repeated application, to the following more comprehensive °Theorem 12. Let R= R (x0, oP, a®@, ..., 39) denote an ex- pression built up, by a finite number of additions, subtractions, multi- plications, and divisions, from the letters x, 2®, ..., x®, and arbitrary numerical coefficients; and let GD, 9), ..., be p given sequences, converging respectively to EV, E®,..., E®). Then the sequence of the mumbers R,=R(®, 2@, ..., e®) RED, £9, .., §0) provided neither in the evaluation of the terms R,, nor in that of the number RED, EP, ..., EW), division by O is anywhere required. These theorems give us all that in required for the formal mani- pulation of convergent sequences: We give a few more Examples. 1. z, — & implies, if a > 0, tnvariably, - 42. a —ab. For at ae = af (a®n=% — 1) ts a null sequence by 35, 3. 2. x, — & implies, if every z, and also & are > 0, that log a, — log §. Proof. We have log x, — log & = log ge = log (1 + Boh which by 88, 7 is a null sequence, since x, > 0 implies nt >-—1. 14 In theorems 3, 11 and 1la, it is sufficient to postulate that the limit of the denominators is # 0, for then the denominators are, from some point onwards in the sequence, necessarily = 0, and only “a finite number of alterations” need be made, to ensure this being the case for all. 15 More shortly: a rational function of the p variables 2, +2, rE x with arbitrary numerical coefficients. 70 Chapter II. Sequences of real numbers. 8. Under the same hypotheses as in 2., we also have, for arbitrary real p, x2 58, Proof. We have 0 = Zp Cre 2, —&\° ecole) ZT, —& é which by 88, 8 is a null sequence, since > —1 and tends to 0 as n-> 00. (This 1s to a certain extent further completed by 33, 4.) Cauchy’s theorem of limits and its generalisations. There is a group of theorems on limits!’ essentially more pro- found than the above, and of great significance for later work, which originated in their simplest form with Cauchy!® and have in recent times been extended in different directions. We have first the simple 43. OTheorem 1. If @g» X45 ---) is a null sequence, then the arith \metic means + oe Z @, = BEnT Eh n=012..., also form a null sequence. Proof. If ¢ is given > 0, then m can be so chosen, that for every n > m we have |, | < 5 - For these #’s, we then have ’ A a en—m lz! < PE +5 2 ntl” Since the numerator of the first fraction on the right hand side now contains a fixed number, we can further determine #,, so that for / n > n, that fraction remains <5 But then, for every n > n,, we have |z,'| 1. Proof. From y,— 17, since all the numbers are positive, we deduce, by 42, 2, that x, = logy, —& =logy. By theorem 2, it follows that Bites x, Bre = log Vy, Va ove Ya = log y, —slog n. By 42, 1, this at once proves the truth of our statement. Examples. 1 It. t= 1. 2 a0, because Lie n n ne 2 2 3 n ” 2. J2=|1 TTF because tk 1+VE+V3 Va + +...4+Vn — 3. ITV2sV bod yn because {7-1 : n 4. Because (1 hol) —¢ (v. 46a in the next §), we have by theorem 3, VEGF GY CE) - EET = 2! eo ia VY nl or, therefore, 1 »— 1 TV =r 3 qt a relation which may also be noted in the form “ymin Ba =e 72 Chapter II. Sequences of real numbers. Essentially more far-reaching, and yet as easily proved, is the following generalisation of Cauchy’s theorems 1 and 2, due to 0. Toepliiz??: OTheorem 4. Let {2 %;5 +43) be a null sequence and suppose the coefficients a,, of the system 0 Ao %, a a a, (A) 20. “zy “ae Xho a, 1 Apo a, n avai el wie a AL ee eC ww ge. satisfy the two conditions: (a) Every column contains a null sequence, i. e. for fixed p => 0 a,,—0 when n— + oo. (b) There exists a constant K, such that the sum of the absolute values of the terms in any one row, i.e., for every mn, the sum | anol +1 aps | +++ +a, | remains < K. — Then the sequence formed by the numbers x, = 20 To o= p17 oe Apo Ty + 22 = Ayn Tn is also a null sequence. Proof. If ¢ is given > 0, determine m so that for every n> m jo, l< To Then for those #’s, |e, | < | @0 To + he + 2% | = By the hypothesis (a), we may now (as m is fixed) choose n, > m, & . so that for every # > n,, we have |a, 2+: + a,,%,.|< 5 Since for these n’s |x,’| is then § 9. The two main criteria. We are now sufficiently prepared to attack the actual problems of convergence. There are two main points of view from which we propose, in what follows, to examine the sequences which come before us. We have above all to consider the Problem A. Is a given sequence (x,) convergent, or definitely or indefinitely divergent? (Briefly: How does the sequence behave with respect to convergence?) — And if a sequence has proved to be convergent, so that the existence of a limiting value is ensured, we have further to consider the Problem B. To what limit & does the sequence (x), recognized to be convergent, tend? A few exam ples may make the significance of these problems clearer: If for instance we are given the sequences (n(vV2 = 1), ( Va — 1) : ((x ag ((x + 34 1 a 14 ered — 2 Bd vee n (Lt +34 27, 2 ), dic. n? log n examination of their construction shows that there are always two (or more) forces which here, so to speak, oppose one another and thereby call forth the variation of the terms. One force tends to increase, §9. The two main criteria. . 7 the other to diminish them, and it is not clear at a glance which of the two will get the upper hand or in what degree this will happen. Every means which enables us to decide the question of convergence or divergence of a given sequence, we call a criterion of convergence or of divergence; these serve, therefore, to solve the problem A. The problem B is in general much more difficult. In fact, we might almost say that it is insoluble, — or else is trivial. The latter, because a convergent sequence (x), by theorem 41, 1, entirely deter- mines its limit &, which may therefore be regarded as “given” by the sequence itself (cf. footnote to 41,1). On account, however, of the boundless complexity and multiplicity of form which sequences show, this conclusion does not seem very satisfactory. We shall wish, rather, not to consider the limit & as “known”, until we have before us a Dedekind section, or still better a nest of intervals, for instance a radix fraction, in particular a decimal fraction. These latter especially are the methods of representing a real number with which we have always been most familiar. If we regard the problem in this light, we TAY call it the question of numerical calculation of the limit. This question, one of great practical significance, is usually in theoretical considerations of very second-rate importance, for from a theoretical point of view, all modes of representation for a real number (nests, sections, sequences, ...) are precisely equivalent. If we observe further, that the representation of a real number by a sequence may be considered as the most general mode of representation, our problem B may be stated in the following form: Problem B’. Two convergent sequences (x, ) and (,) are given, — how may we determine whether or not both define the same limit, or whether or not the two limits stand in a simple relation to one another? A few examples will serve to illustrate the kind of question referred to: l. let 2 1 Y ’ ( 2 15. = en d = — 2 Lp fie und z, 14+ Both sequences are quite easily (v. 46a and 111) seen to be convergent, But it is not so apparent that if & denotes the limit of the first sequence, that of the second is = &2. 2. Given the sequence Vo 7 17 41 122 B00. in which the numerator of each fraction is formed by adding twice the nume- rator of the last fraction preceding to the numerator of the last fraction but one (e. g. 41 =2.17 +417), and similarly for the denominators. — The question of 1 Numerical calculation of a real number = representation of that num- ber by a decimal fraction. For further details, see chapter VIII 4 46. 7 Chapter II. Sequences of real numbers. convergence again gives no trouble, nor does the numerical evaluation of the limit, — but how are we to recognise that this limit = y2: 8. Let - ( 1 1 1 (— DE To P= lvytr raid Ta r=12,. and let zn be the perimeter of the regular polygon with » sides inscribed in the circle of radius 1. Here also both sequences are easily seen to be con- vergent. If & and &’ are their limits, — how does one see that here & = 8.5? These examples make it seem sufficiently probable, that Problem B or B’ is considerably harder to attack than Problem A. We therefore confine our attention in the first instance entirely to the latter, and to begin with make ourselves acquainted with two criteria, from which all others may be deduced. First main criterion (for monotone sequences). A monotone bounded sequence is invariably convergent; a mono- tone sequence which is not bounded is always definitely divergent. (Or, therefore: A monotone sequence always behaves definitely, and is then and only then convergent, when it is bounded, and then and only then divergent, when it is not bounded. In the latter case the diver- gence is towards -}- 00 or — co according as the monotone sequence is ascending or descending.) Proof. a) Let the sequence (,) be monotone ascending and not bounded. Since it is then (because x, > =,) certainly bounded on the left, it cannot be bounded on the right; given any arbitrary (large) positive number G, there is then always an index #,, for which ge But then, since the sequence is monotone increasing, we have for every n > #n,, a fortiori, x, > G, and so, by Definition 40, 2, actually x, — - 00. Interchanging right and left, we see in the same way that a monotone descending sequence which is not bounded must diverge to — co. Thus the second part of the proposition is also proved. b) Now let (x,) be a monotone ascending, -but bounded sequence. There is then a number K, such that |, | ZL K for every un, so that rz, {ez n,, but to the right of J, lie no more x's. If £ is now the number determined by the nest (J), it can at once be shewn that x — &. For if ¢is given > 0, choose the index p so that the length of L is less than z. For # > Ny» all the 2,8 lie, together with &, in I so that for these u's we must have lz, — &| 0, chose m > 2 G; then for n> 2™ felt erin +l Sr 1 1 m—1 = >atart+iy Lisl, L+. ese my Sg iG, The sequence is thevefore not bounded and consequently diverges — + OO. 80 Chapter II. Sequences of real numbers. — 4. If 0 =(x,|y,) is an arbitrary nest of intervals, the left and right end- points of the intervals respectively form two monotone, bounded and therefore convergent sequences. We then have lima, =limy,=(@,|7,)=0- 46a. As a particularly important example, we will consider the twc sequences whose terms are 1 z,= (1+ 1)" and y= fri . {n=1,2:3,..4) We have no means of perceiving immediately (cf. the general remark on p. 77) how the sequences behave as # increases. We proceed to show first that the second sequence is monotone descending, that is to say that for n > 2 fob hie lp Wp IT n—1 n This inequality is in fact equivalent to 1 n Ita 1 Eat 14 % n or to n? ; 1 Y 1 as > 13 ie io +t, Fl-L=, But the truth of this inequality is evident, since, by Bernoulli's in- equality 10, 7 we have, for « > — 1, «== 0 and every un > 1, +e)" >14na, or in particular 1 n n n 1 pat Yen het ty As, moreover, y, > 1 for every n, the sequence (y,) is monotone des- cending and bounded, and therefore convergent. Its limit will ofter occur later on; itis, since Euler's time, denoted by the special letter e®. As regards this number, we can only deduce for the present that 15e 1, n 1 (ten >1—t=1—-. The sequence (x) is therefore monotone increasing. As, in any case, j Tl A we have, for every un, x, 0, a number ng = n,(e) can always be assigned, such that for any two indices n and n' both greater than n,, wz have in every case [2 — ac, | x8, —— We first give a few ; Explanations and Examples. 1. The remarks 10, 1, 3, 4 and 9 are also substantially applicable here; and the reader is recommended to read them through once more in this con- nection. 2. The criterion states — to put it in intuitive language: all z,’s with very high indices must lie very close together. [spi z,=0, #,=1, and let every term after these be the arithmetic me etween the two terms which precede it, i. e. for n = 2 Xp _1T%n_g 2 so that z,=1, x; =3%, 2, =3%,.... In this evidently no! monotone sequence it is clear, on the one hand, that the differences between consecutive terms form a null sequence; for it may be verified quite easily by induction that = 1y 4 Tat1— Ta = on - Tp = and so tends to 0. On the other hand, between these two consecutive numbers all the following ones lie. If therefore, after ¢ has been assigned > 0, we choose p so large that x < &, we have Tp — o, | = 8 provided only »# and »’ are >>. By the 2° main criterion the sequence (z,) is thervefore convergent. The limit & also happens to be easily obtainable. A little reflection in fact leads to the surmise that & =%. In point of fact, the formula 9-2 f= Ty? :% ree ow can immediately be proved by induction and shows that z, — 3% is actually a null sequence. Before trying to fathom the meaning of the 22¢ main criterion further, we proceed to give its Proof. a) That the condition of the ont — let us call it for brevity its ¢-condition — is necessary, i. e. that it is always fulfilled Tk41— Zk Typ —Th—1 hi it follows that if proved for every n < k, it is true for n =%k+41. 4 This is true for n=0 and 1. From a; ;,—;4,= § 9. The two main criteria. 83 if (x) is convergent, is seen thus: If x, —£, then (#, — &) is a null sequence; given & > 0, we can so choose n, that for every n>, jo — Eis < a: If besides n, we also have un’ > #n,, then |x, — &| is also A and so 2, — 2p | = (0, — 8 ~ (ay —H| |v, — | -]ow — &] 0 being chosen, |x, — x, | must always be < ¢ provided only the indices # and #’ both exceed some sufficiently large value. If we suppose the one fixed and denote it by p, then we may also say: Given any & > 0, we can always assign an index p (actually, as far to the right as we please) so that for every n > p lo, =m, |< 2. 1 1 1 9 Zz’ isis y 9% site 1) There is an index p, such that If we choose successively &= . then we get: 1 for every uw > §,, we have a —u, | Eo 2) There is an index p,, which we may assume > p,, such that 1 for every n > p,, we have |x, —ua, |<, and so on. A kth step of this kind gives: k) There is an index p,, which we may assume > Pry» such that for every n> p,, we have |x, — Tp | =< Ls Accordingly we form the intervals J: 1. The interval x, — 5... ®, 4 § call J; it contains all the zs for mw > p,, in particular, therefore, the point x, . It therefore contains in whole or part the interval @,, — 1... , +1, in which all zs with # > p, lie. As these points also lie in J, they lie in the common part of the lwo intervals. This common part we denote 2) by J, and may state: J, lies in J, and contains all points x, with # > ,. If in this result we replace p, and p, by p,_, and Dor and denote therefore % “es bn which lies in Juzy» We may then state: J. lies in J... and contains all points z, with n > p,. k) by J, the portion of the interval xp — 48. 84 : Chapter 1I. Sequences of real numbers. But (J,) is then a nest of intervals; for each interval lies in the preceding and the length of IL i= or Now if £ is the number thus determined, we assert, finally, that z,—£. In fact, if an arbitrary ¢ > 0 be now given, we choose an index 7 so large that Z< eg. We then have for every n> p,, |x,—&|is p,, lies in J and the length of J. is n: Tp — — x : EEL 2 } ) ” nT | ET ow 11 If inside the bracket, we take the successive terms in pairs, we see (cf. later 81 ¢, 3) that the value of the bracket is positive, so that 1 1 1 ol Ln Ee If we now let the first term stand by itself and take the following terms in pairs, we see further that [a= l= 1 TE Therefore |,’ —x,| is <¢, provided » and #’ are both > +. The sequence & [2p — 2p | < 5—— is therefore convergent. 2. a, = (+1 et 1, we have already seen in 46, 3 that (z,) 1s not convergent. With the aid of the. 20d main criterion, ie is deducible from the fact that here the &- condition is rot satisfied for std For however 7, may be chosen, we have for et and »’ =2# (also as > ny) 1 1 1 n41 Fat tym Pon" 3 not therefore 0 can n, be so assigned that the e-condition is then fulfilled; there exists on the contrary (at least) one particular number g >> 0 such that, l= Tn" 5 We shall become acquainted with other proofs of this fundamental cri- terion. The proof given above leads immediately to the definition of the limit by the aid of a nest of intervals. — A critical account of earlier proofs of the criterion may be found in A. Pringsheim (Sitzungsber. d. Akad. Miinchen, Vol. 27, p. 303. 1897). § 9. The two main criteria. 85 above every number ny, however large (therefore infinitely often) two positive in- tegers » and »’ may be found for which | %0 — 2a | Z£>0. 4. The 2°9 main criterion is now usually, after P. du Bois Reymond (Allge- meine Funktionentheorie, Tiibingen 1882), called the general principle of conver- gence. In substance, it originated with B. Bolzano (1817, cf. O. Stolz, Mathem. Ann. Vol. 18, 1881, p. 259) but was first made a starting point, as an expressly formulated principle, by 4. L. Cauchy (Analyse algébrique, p. 125). Our main criterion may also be given somewhat different forms, which are sometimes more convenient in applications. We suppose the notation for the numbers # and #’ so chosen that #’ > #, and therefore we may write #’ = » -}- k, where £ is again a positive integer. We then formulate thus the © Second main criterion (Form 1a). 49. The necessary and sufficient condition for the convergence of the sequence (x) 1s that, given any ¢ > 0, a number ny =n, (e) can always be assigned so that for every n> mn, and every, k > 1 we always have |p <8 From this statement of the criterion we can draw further con- clusions. If we suppose quite arbitrary natural numbers oir Rags sovs soos chosen, then we must have, in view of the above, for every n > 7, | Tn tien — 2, | 0, there certainly exists, if (2) is convergent, (v. Form 1a) a number m, such that for every nu > m and every k >1, we have | Zp — 2, | mn,, we have always », > m. But then, by the preceding, we have, for n > n,, always |, +x, = z, | =|d,|> 0. This could not then be a null sequence. ? Equal or unequal, monotone or not monotone. § 10. Limiting points and upper and lower limits. 87 § 10. Limiting points and upper and lower limits. The concept of the convergence of a sequence of numbers as defined in the two preceding paragraphs admits of another, some- what more general mode of treatment, by which we shall at the same time become acquainted with some other concepts, of the utmost importance for all that comes after. In 39, 6, we have already illustrated the fact of a given sequence (z,) being convergent by saying that every e-neighbourhood (however small) of £ must contain all the terms of the sequence — with the possible exception of a finite number at most. — There is therefore in every neighbourhood of &, however small, certainly an infinite number of terms of the sequence. For this reason, £ may be called a limiting point or point of accumulation of the given sequence. Such points may, as we shall at once see, occur also in the case of divergent sequences, and we define therefore quite generally: ODefinition. A number & shall be called a limiting point* of a given sequence (x,) if every neighbourhood of &, however small, contains an infinite number of the terms of the sequence; or, therefore, if, for any chosen & > 0, there is always an infinite number of indices n for which le, —&| 2, there is obviously only a finite number of positive fractions for which the sum of numerator and denominator =p, namely the fractions Et Ls rie Se Of these we suppose left out all those which are not in their lowest terms, and now consider in succession all the fractions thus formed for p =2, 3,4, .... This gives the sequence, beginning with 3.2 1 (a) 1,2; 2.5704 SE Ba RY which contains all positive rational numbers. 1f after each of these numbers we insert the same number with sign changed and start with 0 as first term, we have in the sequence * German: Hdiiufungswert, Hiufungspunkt or Haufungsstelle. (Tr.) 88 : Chapter II. Sequences of real numbers, 1 1 1 1 (b) 0,1,~1,3,'=2, 5, =~, 5, 8, w=, "=, 4, =4, 588 20 mm R E Ld ci RS RA thus formed obviously all rational numbers occurring, each exactly once. For this remarkable sequence every real number is a limiting point; for every neighbourhood of every real number contains an infinity of rational numbers (cf. p. 11). 4. We shall frequently make use of the principle of arrangement in order applied in this example. We therefore formulate it somewhat more generally: Suppose that for every %k of the series 2=0, 1, 2, ... a sequence z®, 2, Godse ¢=012..) is given. We can then, in many different ways, form a sequence (x,) which con- tains every term of each of these sequences and contains it exactly once. The proof consists simply in assigning a sequence (x,) which fulfils what is required. For this purpose we write the given sequences in rows one be- low the other: : =, »®, «9, ey i 1 1 1 of) ale oll at, . - . . x. . . . . . . k ) k ol ofl al Ls tell The “diagonal” of this system which joins the element 2 to the element al then contains all elements wih for which 24+ #n =p, and no others. They are fp +1 in number. These terms we write down in succession, taking »=0,1,2, ..., and describe each of the diagonals say from bottom to top. Thus we obtain the sequence 1) 2 0 20, a2, ), 2 1 3 2 a0) 900 GoD, which evidently fulfils the requirements. (drrangement by diagonals¥*). Another arrangement frequently used is that “by squares”. Here we first write the elements =, a J oe al? of the p*™ row, then the elements standing vertically above zi) in the above system: Fo, ors, 230, These groups of 2p + 1 terms are then written down in succession for p=0, 1, 2, ..., and this gives, beginning with (0) ) 0) xb, =i, 2} 2 2 2 1 0 3 A", a Ray neo ol the arrangement by squares **. If some or all of the rows in the above system consist of only a finite number of terms, or if the system consists of only a finite number of rows; then the arrangements described above undergo slight and immediately ob vious modifications. * German: Anordnung nach Schriglinien. (Tr.) ¥* German: Anordnung nach Quadvaten. (Tr.) § 10. Limiting points and upper and lower limits. 89 5. An example similar to 3. is the following: For every p = 2 there are exactly p —1 numbers of the form Spl for which the sum of the positive integers k and m is equal to p. If we suppose these written down in succession, for p=2, 3,4, ..., we obtain the sequence 2.8 faa Salt a: oY a? i 5° 1’ 6 6: "00 8 1.) 9 a2 1° * 0 2, We find that this sequence has the limiting points 0, 1, and no others. 6. As in the case of the limit of a convergent sequence, the limiting points of an arbitrary sequence may very well not belong to the sequence itself. Thus in 3. the irrational numbers, and in 5. the value 0, certainly do not belong to the sequence concerned. On the other hand, in both cases the value 1, for instance, is both a limiting point and a term of the sequence. We proceed to give a theorem which is fundamental for our purpose, due originally to B. Bolzano®, though its significance was first fully recognised by K. Weierstrass®. OTheorem. Every bounded sequence possesses at least ome limit- 54. ing point. Proof. We again determine the number in question by a suitable nest of intervals. By hypothesis there exists an interval J; which contains all the terms of the given sequence (z,). To this interval we apply the method of successive bisection and designate as J, its left or right half according as the left half contains am infinite number of the terms of the sequence or mot. By the same rule we designate a definite half of J, as J,, and so on. Then the intervals of the nest (J) so formed all have the property that an infinite number of terms is contained gn each, whilst to the left of their left endpoint there is always at most a finite number of points of the sequence. The point & thus defined is obviously a limiting point; for if ¢ > 0 is given arbitrarily, choose from the succession of inter- vals J, one, say J, whose length is 0, we have, for an infinite number of indices, |@, — &| < e, we have, in particular, for a suitable n =k, | 7, — &| < 1; for a suitable n =k, > &,, we have similarly | z;, — &| < 1, and in general, for a suitable n = £&, > k,_1 : lo, =< t = 2,5..J For the subsequence (x)= (x,) thus picked out, we have z/—&, as (23, — &), by 26, 2, forms a null sequence. The proof of the theorem of Bolzano-Weierstrass gives occasion for a further most important remark: The intervals J of the nest there constructed not only had the property that within them lay an infinite number of terms of the sequence (x), but as we noticed, they had the further property that to the left of the left endpoint of any definite one of the intervals there lay always a finite number only of the terms of the sequence. From this, however, it follows at once that no further limiting point can lie to the left of the limiting point & already determined. For if we choose any real number §&'-< &, we have &= 3 (§ — &’) < 0; choosing an interval J of length » « n> 10 Or by reflection at the origin. 11 These theorems are again obvious except in the case in which the sequence (x,) has an infinite number of limiting points, like e. g. the sequence 33, 5. For among a finite number of values these must ipso facto be both a greatest and a least. * The German text has ‘untere Hdaufungsgrvenze, unieres Limes, Limes inferior”; and “obere Hdaufungsgrenze, oberes Limes, Limes superior’. (Tr.) . Lah aad to At § 10. Limiting points and upper and lower limits. 91 (possibly omitting the subscript n—o00). If pu is the greatest li- miting point of the sequence, we write lime, = u or limsupz, =u n—>w n-—>®w and call u* the upper limit or limes superior of the sequence (x). We have necessarily always » 0, we have still for an infinite number of n’s, 2, p—0) but for at most a finite number of w’s, x, p-tet Before we give a few examples and explanations of this theorem, let us complete our definitions for the case of unbounded sequences. Definitions. 1. If a sequence is unbounded on the left, then we will say that — oo is a limiting point of the sequence; and if it is unbounded on the right, we will say that oo is a limiting point of the sequence. In these cases, however large we choose the number G > 0, the sequence has an infinity of terms below — G or above + G12. 2. If therefore the sequence (x,) is umbounded on the left, then — 00 is ipso facto the least limiting point, so that we have to write z=Ilmz = — co. n->+ xo Similarly we have to write n= lim x, = +00 n>+® if the sequence is unbounded on the right. In these cases, however large we choose the number G > 0, we have, for an infinity of indices, 2, <—Goruw>+0G. : * The German text has “wuntere Hdaufungsgrenze, unteves Limes, Limes inferior’; and “obeve Hdaufungsgrenze, obeves Limes, Limes superior’. (Tr.) 12 Or: There is an index n, from and after which we never have x, p+¢) but beyond every index zn, there is always another »n for which p< +e (> p—s) 13 Here therefore — and similarly in the following definitions — the portion of the straight line to the right of 4 G plays the part of an g-neigh- bourhood of + oo, the portion to the left of — G that of an ¢-neighbourhood of —o0. : 99. 60. > 61. 992 Chapter II. Sequences of real numbers. 3. If, finally, the sequence is bounded on the left, but not on the right and (besides + oo) has no other limiting point, then -}- co is not only its greatest, but at the same time its least limiting point, and we shall therefore equate the lower limit also to + oo: % =limz, = + 00; n>+w and correspondingly we shall have to equate the upper limit to — oo, uw -— limz, = — 0 n—>+ wo if the sequence is bounded on the right, but mot on the left and (be- sides — oo) has no other limiting point. The former (latter) case occurs if and only if, given any G > 0, the inequality z,>6 (@,<—0) holds for an infinite number of w’s, but the 1mequality 5 =0 for at most a finite number of ns. Examples and explanations. 1. In consequence of the preceding definitions, every sequence of numbers now of itself defines, absolutely uniquely, two determinate symbols x and J (which may now, it is true, stand for 4-00 or —oo, and which bear the re- lation x < pw !* to one another. And the following examples show that x and u may actually assume all finite or infinite values compatible with the in- equality x < u. Ta fact, for the sequence we have Ga, x= | u= 1 m=12 Bd idee +o | + 8. tv DY ymall, o42 ot 1, a-t4 ... a + oo 8.2,0,0,b,a0D,-... (a at Rs Bis evident] = n yt ’ 9° 8? 4°? 5 yu -ntly 14 We say of every real number that it is <4 oo and > — oo, and for ‘this reason we occasionally designate it expressly as “finite”. § 10. Limiting points and upper and lower limits. 93 #=—1, p=+1, and both to the left of x and to the right of p lies an infinite number of terms of the sequence (and between » and pu lies no term of the sequencel). It is therefore not at all necessary that there should be only a finite number of terms of the sequence outside the interval » ... ux. Theorem 59 only asserts in fact that at most a finite number of terms of the sequence can lie to the left of » —& or to the right of ute. 3. “A finite number of alterations” has no effect on the limiting points of a sequence — none, in particular, on its upper and lower limits. These therefore represent an ultimate property of the sequence. 4. Since a sequence (x,) determines both the numbers x» and u with complete uniqueness, and since their value, in connection with our definition, was also enclosed by a well defined nest of intervals, we have herein a new legi- timate means of defining (determining, giving) real numbers: a real number shall henceforth also be regarded as ‘given’, if it is the upper ov lower limit of a given sequence. This means of determining real numbers is evidently still more general than the one mentioned in 41, 1 since now the sequence utilised need not even be convergent, or be subject to any restriction whatever. As may be seen, in the light of 85, we have also the following Theorem. The upper limit wu of the sequence (v,), u= lima, , is also, in the case pu == + 00, characterised by the two following conditions: a) the limit &' of every convergent sub-sequence (x) of (x,) is invariably < pu; but there exists b) at least one such sub-sequence, whose limit is equal to pu; — and correspondingly for the lower limit. A concept related to that of the upper and lower limits, though one which must be sharply distinguished from it, is the concept of upper and lower bounds of a sequence (x), which is derived from the following consideration: If no term of the sequence lies to the right of p=limz,, so that for every m, x, < nu, then pu is a bound above (8, 4) of the sequence, — but one which cannot be replaced by any smaller one; u is therefore in this case the least bound above. But such a least bound also exists if there is a term of the sequence => p- For if for instance z, is > u, then by 859 there is certainly only a finite number of terms in the sequence which are > , and among these there is necessarily (8, 5) a largest one, say x,. We then have, for every n, ©, Sw, i. e. x is a bound above of the se: quence, — but again one, which cannot be replaced by any smaller one. Every sequence bounded on the right therefore possesses a definite least bound above. Since, in the same way, every sequence bounded 15 Whereas therefore a nest of intervals (with rational endpoints) was at first to count as the only means of defining a real number, we have now deduced quite a series of other means which we now admit as equally legi- timate: Radix fractions, Dedekind sections, nests of intervals with arbitrary real endpoints, convergent sequences, upper and lower limits of a sequence. In all these cases, however, we saw how at once to assign a nest of intervals (with rational endpoints) which encloses the given number. 62. 63. 94 ; Chapter II. Sequences of real numbers. on the left must have a definite greatest bound below, we are justi: fied in the following Definition. We define as the upper bound* of a sequence bounded on the right the least of its bounds above (invariably deter- minate by our preliminary remarks), and similarly as the lower bound*) of a sequence bounded on the left the greatest of its bounds below. A sequence unbounded on the right is said to possess the upper bound + oo, one unbounded on the left, to possess the lower bound — oo. The concepts of upper and lower limits are due to A. L. Cauchy (Analyse algébrique, p. 132. Paris 1821) but were first made generally known by P. du Bois-Reymond (Allgemeine Funktionentheorie, Tiibingen 1882). Both nomen- clature and notation have remained variable up to the present day. The parti- cularly convenient notation lim and lim used in the text was introduced by A. Prings- heim (Sitzungsber. d. Akad. zu Miinchen vol. 28, p. 62. 1898), to whom the designations of upper and lower limits are also due**, The previous investigations of this paragraph were carried out quite independently of the considerations on convergence of §§ 8 and 9, and give us, for this very reason, a new means of attacking the problem of convergence A of § 9. It may be shewn that the knowledge of the lower and upper limits x and u of a sequence — the knowledge, therefore, of two numbers whose existence is a priori ensured — en- tirely suffices to decide whether or how the sequence converges or diverges. We have in fact the theorems : Theorem 1. The sequence (x,) is convergent if and only if its lower and upper limits x and pu are equal and finite. If 1 is the common value (different, therefore, from 4-00 or — oo) of » and pu, “then z,— 1. Proof. a) Let =u and their common value = 4. Then, by 59, given g, there is at most a finite number of #’s for which ZX, u-te=41-4e. * German: Obere, untere Grenze (frontier). The word “frontier” is not usual in English writings, though sometimes found in French. The distinction between any bounds and the narrowest bounds is emphasized chiefly by the article the in the latter case; the upper bound and the lower bound always de- noting the latter. For fear of ambiguity, however, the word “bound” in the general sense is avoided as much as possible in English text-books. (Tr.) ** We have omitted reference here to the untranslated term ‘‘Hdufungs- grenze” of the German text: “Die im Texte benutzte ausfiihrlichere Bezeich- nung Haufungsgrenze soll nur den Unterschied zu der soeben definierten unteren und oberen Grenze stirker betonen”. (Tr.) - § 10. Limiting points and upper and lower limits. 95 For every m > some n,, we therefore have A—e 0, we have, for every n> m,(¢), A —¢e A—¢ is satisfied for an infinite number of #’s, but the inequality vr, A1+¢) for at most a finite number of n’s. The former inequalities (with <) imply » = 2, the latter u == 1. This proves all that we required. Theorem 2. The sequence (x,) is definitely divergent if, and only if, its upper and lower limits are equal, but have the common value + oo or —oo'% (And in the former case it diverges of course to + oo, in the latter to — co.) Proof. a) If x=pu = +4 00 (or — 00), then this signifies, by 60, 2 and 3, that, given G > 0, we have from and after a certain #, we therefore then have limz = +4 co (— 0). b) If, conversely, limx, = + oo, then, given G > 0, we have for every n after a certain ny, xz, > -} G; therefore the inequality x, << 4- G is satisfied for at most a finite number of #’s, whereas the inequality x, > -}- G is satisfied for an infinite number of #’s. But this implies, by 60, that x = + co and ipso facto also yu = oo. Therefore ¥ = yu = +4 00. And in precisely the same way we show that if limx, = — oo, then » = yu = — oo. Theorem 3. The sequence (x,) is indefinitely divergent if and only if its upper and lower limits are distinct. Proof. a) If x=, so that x» < u, then we are faced with neither the situation in Theorem 1, nor that in Theorem 2; therefore (=) must necessarily diverge indefinitely, And in the same way: b) If (z,) is indefinitely divergent, then we are faced with neither the situation in the first, nor in the second Theorem, and therefore we cannot have x = u. 16 In occasionally speaking of the symbols + oo and — oo (which are certainly not numbers) as “values”, we make use of a mere verbal licence, to which no importance should be attached. 64. 65. 96 Chapter Il. Sequences of real numbers. The content of these three theorems provides us with the following Third main criterion for the convergence or divergence of a se- quence: The sequence (x,) behaves definitely oz indefinitely, according as its upper and lower limits are equal or distinct. In the case of de- finite behaviour, it is convergent or divergent, according as the _ common value of the upper and lower limits is finite or infinite. The following table gives a summary of possibilities as regards the convergence or divergence of a sequence and of the designations used in this connection. #=u, both =1= 1 oO x=p=-+00 or —00 Zl convergent (with limit 1) | divergent (or possibly: con- Gm oad vergent) towards (or: with n= Gh %) limit) 4-00 or — 00; in both indefinitely cases: definitely divergent. divergent Tp —> A lima, =+400 or — © (for #— + 00) Z,—-+} 00 or — 00 ~ convergent divergent definite behaviour Indetinite behaviour Finally we proceed to prove, — as an application, also, of the concepts newly introduced, — a somewhat more complicated theorem of limits which will be useful to us later (§ 60) and offers to a certain extent a substitute for the missing converse to Cauchy’s theorem 433, 2 or its generalisation 44, 2; the notation corresponds to that of the latter statement. 3 Theorem. If o,, 0, -.., 0, ... are any positive numbers such y that the sum +a t-+a, 1-00 when n—>00, and if «> 0, — then _ Cox +o, X41 0p Xp 2 on, + - SBT tn, implies that b) x, -_— & . Proof. We begin with the following considerations: 1. If (x,) converges, say — &, then by 44, 2 the values x! = Glo teTit---ta 2, n Cyto tend to the same value &. By a) we must then have & = ¢&. 2. It suffices, therefore, to prove the existence of limz, , — or even the existence of lim 2,’ only, since by a) either both or neither exist. § 10. Limiting points and upper and lower limits. 97 3. The case xz, — + co, — and therefore by 44, 2a, x'— 4-00, — certainly, by a), does not come into account. For in this case x, would, infinitely often, be greater than all preceding x,’s, and in parti- cular, also infinitely often, xz, > x 7, or, also, «x, += (1 = «) x, = «(, x, + mt > x, In contradiction with the hypothesis a), the numbers on the left of this inequality must then be capable of assuming arbitrarily large values. — Similarly #, cannot — — 00. 4. In consequence of a), we therefore either have, as asserted, x, — &, or else (x,) is indefinitely divergent, which must then also be the case with (xz). It only remains to show that the latter possibility is incompatible with the hypothesis a). For this purpose, we make the following statements: L If, for a particular n, / / ’ x, 2, ri ig In fact, , 9: Let p be an index for which the first inequality holds, and g > p one for which the second holds: z, r< ut, x, !'~ 9. Now if it so happens iy yl occurs for n=4q, 1. e x, < x) then by I we have 24_1 > =, I ~>v. HI 50 happens that case I occurs also for n= q — 1, then we Shave Ton > 2). Ne > 9, and so on. How- ever, for all the values n==4q, g=1, :. 2s 2, p+ 1, this case can not occur, since we could then in tha same way deduce x, = 66. terms we now denote by s 67. 98 Chapter II. Sequences of real numbers. in contradiction with the meaning of p. Among these numbers g, g—1,..., p+ 1 there is therefore a first, say m, for which case I does not occur, so that x, > x, and a, > api >>) > 0, and therefore, in any case, = Ts But then, as « is assumed > 0, we also have en, + — duh, =u, + ale, — ay) 2 oh > 0. If, therefore, (z,’) is indefinitely divergent, this inequality is satisfied for an infinite number of indices m. And since in this case we can show, in exactly the same way, using statement II, that for an infinite number of indices m, we must have er, +1—a)z, 0 and this new symbol we call an infini’e series; the numbers s are called the partial sums or sections* of the series. — We may there- fore state the ODefinition. An infinite series is a new symbol for a definite 68. sequence of mumbers deducible from it, namely the sequence of its par- tial sums. Remarks and Examples. 1. The symbols © © ® Bo > Cn ata Xa,; ata +---+a,+ > an n=2 n=1 n=m+1 0 shall be entirely equivalent to 3 a,. The irdex = is called the index of sum- n=0 mation. Of course any other letter may take its place a0 0 >a; ata tata, etc r=0 0=3 The numbers a, are the ferms of the series. They need not be indexed from 0 onwards. Thus the symbol x a, denotes the sequence (a,, a, ay, a,+a;,+a;,...) 2=1 and more generally, wn 2 ax k=p denotes the sequence of numbers s,, s,41,, Sp4q,... given by Sp=ly TU ta, for n=p,p-t1,.., Here p may be any integer Bo; Finally we also write quite shortly 2 ay when there is no ambiguity as to the values which the index of summation has to assume, — or when this is a matter of indifference. 2. Forn==0,1,2,... let a, be 1 1 8) =i 0) “hrhmto’ 9=C0 pecs @=c1rent: nt-1" ) ’ — a—— 1 . - (etn) (etntl)’ c) =]; d) =; h) «= a real number +0, —1, —2, ... * German: Teilsummen oder Abschwitte. 100 Chapter II. Sequences of real numbers. We are then concerned with the infinite series n=0 2" = 1 1. vg BD SEITE Trt att Pll ti pei HOLILILDL. 3 J Seay ply DIS r Ep Ln 1 1 1 1 BY THT IN CIN Ine And we have in these simply a new — and as will be seen, very con venient — symbol for the sequences (s,, $;, Sp. ...) for which s, is Lad 1 1 a) = lip reel 20 1 1 1 Beratost Le tT EF DED 1 1 Ds. (olf ey -(1-5)+ (3-3 Sr orl alr n(n41 c) =n+1; g=ritl, oy 1 —1y : €) glegty=te tid (cf. 45,3 and 48,1); fH =31-(=1H7 g) =(= 1 (+1); 1 il 1 5) EL NGS TED are) (ar) (c-) + a i 3. We emphasise above all that the new symbols have no significance in themselves. Addition, it is true, is a well-defined operation, always possible, with regard to two or any particular number of values, in one and only one way. The partial sums s, therefore, however the terms a, may be given, have ow under all circumstances definite values. But the symbol 3} a, has in itself no n=0 meaning whatever, — not even in a case as transparent, seemingly, as 2a; for the addition of an infinite number of terms is something quite undefined, some- thing perfectly meaningless. It must be considered substantially as a con- ‘vention that we are to take the new symbol to mean the sequence of its partial sums. 17 I. e. equal to 1 or 0, according as » is even or odd. § 11. Infinite series, infinite products, infinite and continued fractions. 101 4, The reader should take particular care to distinguish a series from a sequence!®: A series is a new symbol for a sequence deducible by a definite rule from it. 5. The symbol with the sign of summation “3}”’ can of course only be used when the terms of the series are formed by an explicitly assigned law, or when a particular notation is available for them. If for instance the numbers 111.77 1 ars 3 TP 1B Wo or the numbers rr 11 1 1 A 3 F188: a8’ 9 95 BY i are to be the terms of a a we shall have to use the explicit symbols 1 il +1 tl il Le and 1 er al lent ielal +o and write down as many terms as necessary, till we may assume that the reader has recognised the law of formation. For the first of these two series, this may be expected after the term 4: the terms are the reciprocals of the successive prime numbers. In the second example it will not be known even after the term #1, how to proceed: the denominators of the terms are meant to be the integers of the form p?—1 {0,9=2,3,4,..) in order of magnitude. We now adopt the further convention that all expressions used to describe the behaviour, in respect of convergence, of a sequence are to be carried over from the sequence (s,) to the infinite series 2'a, itself. Thereby we obtain in particular the following Definition. An infinite series Xa, is said to be convergent, definitely divergent or indefinitely divergent, according as the sequence of its partial sums shows the behaviour indicated by those names. If, in the case of convergence, s,—s, then we say that s 1s the value or the sum of the convergent infinite series and we write for brevity 2 =5" v=0 in the case of definite divergence of (s,), we also say that the series is definitely divergent and that it diverges to + co or — oo according as s,— +00 or — — oo. If finally, in the case of indefinite diver- gence of (s,), » and w are the lower and upper limits of the sequence, then we also say that the series is indefinitely divergent and oscillates between the (lower and upper) limits x and u. 18 The additional epithet of “infinite” may be omitted when obvious. 1% Exactly as we may now, in accordance with the footnote to 41, 1, write (s,) =s. 102 Chapter 1I. Sequences of real numbers. Remarks and examples. 1. It is at once obvious that the series 68, 2a, b and h converge and have for sums +2, 1 and 31 respectively; 2c and d are definitely divergent towards o +00; 2e is convergent and has for sum the number s defined by the nest (Sag, | S25) 20. 2t, finally, oscillates between 0 and 1 and 2g between — and +00. . 2. As regards the term sum the reader must be expressly cautioned about a possible misunderstanding: The number s is not a sum in any sense previously in use, but only the limit of an infinite sequence of sums; the equation 0 Da,==S or ata, +--+a,+---=s n=0 is therefore neither more nor less than another way of writing lims,=s or s,—s. It would therefore seem more appropriate to speak not of the sum but of the limit or value of the series. However the term “sum” has remained in use from the time when infinite series first appeared in mathematical science and when no one had a clear notion of the underlying limiting processes or generally, of the “infinite” at all. 3. The number s is therefore no sum, but is only so named, for the sake of brevity. In particular, calculations involving series will in no wise obey all the rules for calculating with sums. Thus for instance in an (actual) sum we may introduce or omit brackets in any manner, so that for instance, - 1-141 ~-1=(~-Dtd~-D=l-I~Y=1=0, * But on the contrary Pri = lt tm n=0 is not the same thing as A-D+A-D+A—D+ +++ =0+0+04... or as 1-0-D-Q=-D—-0 == =1=0—-0-0-".%, Nevertheless, calculations involving series will ‘have many analogies with those involving (actual) sums. The existence of such an analogy has, however, in every particular case to be first established. 4. It is also, perhaps, not superfluous to remark that it is really quite 1 raradoxical that an infinite series, say 2 gus should possess anything at all n=0 1 1 1 1 1 1 20 In fact spam{iz Lie ll—lY poe tn {ly 0) = 1 1 ai tot tern so that s; s,>s,>.... Finally Sop Sap—1 = tg i. e. positive and tending to 0. By 46, 4 and 41, 5, we have s, — (S;;._; | S33). Cf. 8le, 3 and 82, 5 where these considerations are generalised. § 11. Infinite series, infinite products, infinite and continued fractions. 103 capable of being called its sum. Let us interpret it in fourth-form fashion by shillings and pence: I give some one first 1 s., then !/,s., then !/, s., then 1; s., and so on. If now I never come to an end with these gifts, the question arises, whether the fortune of the recipient must thereby necessarily increase beyond all bounds, or not. At first one has the feeling that the former must occur; for if I continue constantly adding something, the sum must — it seems — ulti- mately exceed every value. In the case under consideration this is not so, since for every = 1 neltlsly cof bent remains << 2. The total gift therefore never reaches even the amount of 2s. And if we ; : : 1: now, in spite of this, say, that >’ 5: equal to 2, then we are really only using an abbreviated expression for the fact that the sequence of partial sums tends to the limit 2. 5. In the case of definite divergence we can also, in an extended sense, speak of a sum of the series, which then has the “value” +o or — oo. Thus for instance the series 21 1 1 1 AES nt att is definitely divergent, and has the “sum” + op, because by 46, 3 its partial sums — 400.28 We write for short which is only another mode of writing for tim (144+ ov +) = + co, n 6. In the case of an indefinitely divergent series however, the word “sum” loses all significance. If in this case lims, =x and lims, =u), then we said, in the above, that the series oscillates between » and uw. But it must be carefully noted (cf. 61, 2), that this refers only to a description of the ultimate behaviour of the series. In fact the partial sums s, need not lie between x and uy. Thus, for instance, if a, = 2, and for » > 0, my alent n n+ 1 we can at once verify that n4 2 = yet ber ht, = 0) ony r=012,..) and therefore lims,=—1, lims, =+1. But all the terms of the sequence (Sn) 2 If therefore the payments discussed in 4. have the values 1s., 1, s., ys. Y,;s.,... the fortune of the recipient now does increase beyond all bounds. It is not at first at all obvious to what it is due that in the case 4, the sum does not exceed a modest amount, whereas in the present case it exceeds every bound. The divergence of this series was discovered by John Bernoulli and published by James Bernoulli in 1689; but seems to have been already known to Leibnitz in 1673. 104 Chapter II. Sequences of real numbers. lie outside the interval —1 ... +4 1, alternately on the left and on the right, so that an infinite number of terms of the sequence lies on both sides of the interval. ! 7. As we emphasized above that a series > a, represents merely the sequence (s,) of its partial sums, — and therefore is merely another mode of symbolising a sequence, so we may easily convince ourselves that conversely every sequence, (z,, Z,, ...) may be written as a series. We need only write B=, = — Ty G=Uy—F),:003 CG =B Wp qyere. BZ1) For then the series Its @® ay =x,+ 3 (X) — Ty) n k=1 has for partial sums So =p, $3 =%5+ @ — x) =2, and generally for n >1 Sa=T+ (8, — Tp) + @ — 2) + +++ + (Fney — Tug) + @n — Tuy) = 7, so that the above written series does actually stand for the sequence (z,). The new symbol of the infinite series is therefore neither more special nor more general than that of the infinite sequence. Its significance resides princi- pally in the fact that the emphasis is on the difference a,=s, —s,_, of each term of the sequence (s,) from the preceding, rather than on these terms themselves. 8. With regard to the History of Infinite Series, an excellent account is given in a little book by R. Reiff (Tiibingen 1889). Here it may suffice to mention the following facts: The first example of an infinite series is usually ascribed to Archimedes (Opera, ed. J. L. Heiberg, Vol. 2, pp. 310 seqq., Leipzig 1913). i ; He, however, merely shows that 1 4 : + eee + Te remains less than 2 whatever value » may have, and that the difference between the two values is oo and consequently less than a given positive number, provided # be taken sufficiently large. A more general use of infinite series does not, however, begin till the second half of the 17% Century, when N. Mercator and W. Brouncker, in 1668, while engaged on the quadrature of the hyperbola, discovered the logarithmic series 120, and when I. Newton, in 1669, in his work De analysi per aequationes numero teyminovum infinitas placed their use on a firmer basis. In the 18™ Century the consideration of principles was, it is true, entirely neglected, but the practice of series, on the other hand, was developed, above all by Euler, in a magnificent manner. In the 19% Century, finally, the theory was established by A. L. Cauchy (Analyse algébrique, Paris 1821) in an irre- proachable manner, except for the want of clearness which then still attached to the concept of number as such. (For further historical remarks, sce Intro- duction to § 59.) II. Infinite products. Here we are concerned with products of the form Uy lat Uso nelly nes OC. JT; they must be taken, in a precisely similar manner to the infinite series just considered, simply as a new symbolic form for the well-defined sequence of the partial products Pr=%; Pa=U Uy; et; Po=UUy...U; ens § 11. Infinite series, infinite products, infinite and continued fractions. 105 However we shall later, with reference to the exceptional part played by the number 0 in multiplication, have to make a few special con- ventions in this connection. (n+ 1) 1. If for instance we have, for every n=>1, u,= PLY then the in- finite product y amy Pn +2) 1-324 3.5 4-6 n (n+ 2) represents the sequence of numbers neti a=ll n=Bh a 8=20ED 2. The additions and remarks just made in I retain mutatis mutandis their significance here. All further details will be considered later (Chapter VII). 111. Infinite continued fractions. Here the sequence (z,) under examination is formed by means of two other sequences (a, a...) and (b,, b;,...), by writing: a a a x, = by, 4 = by 22, Tg =Dy+—"—, a=b+ : s 1 and so on, z,, in the general case, being deduced from a, ; by substituting 3 a, for the last denominator b,_, of x,_, the value bay +", and proceeding thus n ad infinitum. For the “infinite continued fraction” so formed there is no uni- versally acknowledged new symbol in use. The most natural notation for it would be 7 @D a b+ A x n n=1 Here also a few special conventions have to be made, to take the fact into account that in division the number ( again plays an exceptional part. The subject of continued fractions we shall not, however, enter into in this treatise 22, Of the three modes of assigning a sequence discussed above, that by infinite series is by far the most important for all applications in higher mathematics. We shall therefore have to deal mainly with these. — Since series merely represent sequences, the introductory developments of § 9 provide us with the points of view from which a given series will have to be investigated: Together with the problem A which concerns the convergence or divergence of a given series, we have again the harder problem B, which relates to the sum of a series already seen to be convergent. And for exactly the same 22 A complete account of their theory and applications is given by O. Perron, Die Lehre von den Kettenbriichen, Leipzig 1913. 106 Chapter II. Sequences of real numbers. reasons as we there explained, the second problem will generally present itself in the form: A series 2'a, is known to be convergent; does its sum coincide with that of any other series or with the limit of any other sequence, or does it stand in any assignable relation to such another sum or limit? *® Since the problem A is the easier and since — in contradistinction to problem B — it admits of a methodical solution, we will proceed in the first place to give our attention to this in detail Exercises on Chapter II. 9. Prove Theorems 15 to 19 of Chapter I by the method indicated in the footnote to 14. 10. Prove in all details that the ordered arrangement, defined by 14 and 15, of the system of all nests of intervals, obeys each of the theorems of order 1. (For this cf. 14, 4 and 15, 2.) 11. Carry out the details of the proof required on p. 31; i. e. prove that the four modes of combining nests of intervals, defined by 16 and 19, obey all the fundamental laws 2. 32. For fixed g, with 0'< p< 1, Z,=nm+1)2—n2—0. 138. For arbitrary positive « and fg, (log log 7) * 550. (log n)¥ 3 V3 % 14. Which of the two numbers oo) and (vz)? is the larger? 23 Thus e. g. the series 141 rely cee tle ... will easily be shown to converge. How do we see that its sum coincides with the number e given by the sequence (1 Fa) Similarly we may very soon convince our- selves of the convergence of the two series ti ely wh Bo: sels and => 1.3 a, But how do we discover that if s and s’ are their sums, smo and 45 =x (i. e. equal to the limit in a third limiting process, which occurs in relation to the circle; cf. pp. 200 and 214)? 24 Tn several of the following exercises, a few of the simplest results with regard to logarithms, and the numbers e¢ and xz, are assumed known, although they are only deduced later on in the text. Exercises on Chanter 1L 107 15. Prove the following limiting relations: so Badeoglo dl 0 [eleR) mere) esol ) [Fine ihe, 2 [arene [footy 9 Ver DeTY em Note that in examples a) to d) a teym by term passage to the limit gives a wrong result, whereas in e) it gives a correct result. 16. Let a be 0, », > 0 and the sequence (z,, 7,,...) defined by the convention that for n = 2 a) p= Va ry a b) ow, Shew that in case a) the sequence tends monotonely to the positive root of xz* —2z--a=0; that in case b) it tends to that of 24x —a=0, but with z, lying alternately to the left and to the right of the limit. 17. Investigate the convergence or divergence of the following sequences: a) z,, x, arbitrary; for every n = 2, z,=1 (@y + Trg); D) Ts #;, +o 55: py, Arbitrary; for every n 2p y= Xp + %y_gt+ 22+ +0,%,_p 3 1 (a ag ..., @, given constants, e. g. all equal to 2); €). @,, », positive; for every n 22, wy = Vi, 2, ,} 2%y—1 Tn—g d) =,, x, arbitrary; for every n>2, u,= ’ ) 02 v1 y: yY = n Tas Tacs 18 If in Ex. 5 c we put, in particular, z,=1, x, = 2, then the limit of the sequence is -VT. 19.) Let Ay, gy vv ey Oy be arbitrary given positive quantities ‘and let writey fdr n=1, 2, ... af tap +. +ap p n =, and Ys, =u, 108 Chapter II. Sequences of real numbers. Show that x, always increases monotonely and if one, say a,, of the given numbers is greater than all the others, then x, — a, as limit. (Hine: First show that 5 ) Sy 2 i 20. Somewhat similarly to last Ex., write Bag no Va, +Vag +--+ +Va, = p Te Sy IA IA © |e IA 57 and (5Vi- at pra and show that x,’ decreases monotonely and — Va, a... a. 21. Divide the interval a... b (0 ——. log b — log a 22. Show that in the case of the general sequence of Ex. 5 Zn C= BE a” a(e—p) 23. Set > 0 and let the sequence (z,) be defined by & Tn— Pine ap=nfl, ger on manta a, oy For what values of x is the sequence convergent? (Answer: If and only if 24. Let limz, =x, limz, = ow, ima, — 2, lim x,’ = u’. What may be said of the position of the limits for the sequences (— 2), (z): (Tn + 22), Ex —2); Enza), (&)2 Discuss all possible cases. 25. Let («,) be bounded and (with the possible exception of a few initial terms) let us put @n) _ Pa log (1 +2) Sw Then (o,) and (8,) have the same upper and lower limits. The same holds if we put On ) 1 Br nlog n r 26. Does Theorem 48, 3 still hold if =0 or =+4+ 00? 7. If the sequences (z,) and (y,) given in 48, 2 and 3 are monotone, then so are the sequences (z,’) and (y,’) mentioned there. n Tolooy log +1 ~ Exercises on Chapter II. 109 Cs. If the sequence (2) is monotone and b, > 0, then the sequence n having nth term : a +ag+--- +a, by +0, + RC is also monotone. 29. We have Mot roy 2 2% by b, Ta Yn provided the limit on the right exists and (a,) and (b,) are null sequences, with (b,) monotone, 80. For positive, monotone ¢,’s, Zy+ 2; + i n+ 1 => £ implies Colo+-Ci 2+. Tat Cov Cir stn : 7 Cy provided ) is bounded and C, —» +4 co. (Here Ch=cy+c, Foor dep) a’ 31. If b,>0, and by+b,+ +++ +b, =B,—>+4 0, and x, > + 00, then B 5 (@n— Ty) > & n implies [2 Sahl de % bo +0, + +0, | 82. For every sequence (z,), we invariably have m= n+ 1 —£. wii (Cf. Theorem 161.) 33. Show that if the coefficients “0 of the Theorem of Toeplitz 43, 5 are positive, then for every sequence (x,) the relation lim z, < lim ,’ < lim a, holds, where #,/ =a, + dy, @, + +++ + App Zn. «0. Pare NI. Foundations of the theory of infinite series. Chapter IIL Series of positive terms. § 12. The first principal criterion and the two comparison tests. In this chapter we shall be concerned exclusively with series, all of whose terms are positive or at least non-negative numbers. If 2g, is such a series, which we shall designate for brevity as a series of positive terms, then, since a, > 0, we have Sn rt + a, => Sp—1> so that the sequence (s,) of partial sums is a monotone increasing sequence. Its behaviour is therefore particularly simple, since it is then determined by the first main criterion 46. This at once provides the following simple and fundamental First principal criterion. A series with positive terms either con- verges or else diverges to 4-00. And it is convergent, if, and only if, its partial sums are bounded’. Before indicating the first applications of this fundamental theorem, we may facilitate its use by the following additional propositions: Theorem 1. If p is any positive integer, then the two series oo oo. apd 3g n=0 n=op converge and diverge together®. 1 Only boundedness on the right (boundedness above) comes into question, since an increasing sequence is invariably bounded on the left. 2 More shortly: We “may” omit an arbitrary initial portion. — For this reason, it is often unnecessary to indicate the limits of summation (between which the index » is made to vary). “- 3.0 § 12. The first principal criterion and the two comparison tests. 111 Proof. The partial sums of both series, which correspond to the same index for the terms of the series, only differ by the constant number {(o, a, +++ + 2,1) and are therefore either bounded, or unbounded, for both series simultaneously. Theorem 2. If Zc, is a convergent series with positive terms, then so is 2y,c,, if the factors y, are any positive, but bounded, numbers®. Proof. If the partial sums of J¢, remain constantly < K and the factors py, < y, then the partial sums of 2'y ¢, obviously remain always < y K, which, by the fundamental criterion, proves the theorem. Theorem 3. If Jd, is a divergent series with positive terms, then so is 20,4, if the factors J, are amy numbers with a positive lower bound 0. a Proof. If G > 0 be arbitrarily chosen, then by hypothesis the partial sums of 2d , from a suitable index onwards, are all > G:4. From the same index onwards, the partial sums of 20, d, are then >G. Thus 4,4 is divergent Both theorems are substantially contained in the following Theorem 4. If the factors «, satisfy the inequalities Ol < ou <, then the two series with positive terms 2a, and 2, a, converge and diverge together. Or otherwise expressed: Two series with positive terms 2a, and Xa, converge and diverge together if two positive numbers ¢/ and o’ can be assigned for which, constantly, (or at least from some nn onwards) n SIF tL in particular therefore if a,/~ a, or, a fortiori, if a,/'~a, (v. 40, 5). Examples and Remarks. 1. If K is a bound above for the partial sums of the series Xa, with positive terms, then the sum s of this series is < K (v. 46, 1). 9. The geometric series. Given a > 0, and the so-called geometric sevies Drm Fut atl Lard i, n=0 we have, if a>1, then s, ># and so (s,) is certainly not bounded; the series = We shall in future usually denote by c¢, the terms of a series assumed convergent, and by d, those of a series assumed divergent. 4 Since, in this formulation of the hypotheses, division by a, occurs, the assumption is of course implied that a, > 0 and never =(0. — Corresponding restrictions should be observed in the more frequent cases in the sequel. v1. 112 Chapter III. Series of positive terms. is therefore in that case divergent. But if a 0. From our fundamental theorem we shall in due course deduce criteria which are more and more special, but are also more and more easy to manipulate and more and more effective. This we shall be enabled to do chiefly by the instrumentality of the two following “comparison tests” *: Comparison test of the 1% Lind. Let 2c, and 2d, be two series with positive terms, already known to be the first convergent, the second divergent. If the terms of a given series 2 a, also with positive terms, satisfy, for every nm >a certain m, a) the condition nS Chy then the series 2a, is also convergent. — If, however, for every n > a certain m, b) we have constantly Hh ds then the series 2a, must also diverge”. Proof. By 70,1, it suffices to establish the convergence or di- @D vergence of 3 a . In case a) the convergence of this series results n=m+1 at once, by 70, 2, from that of Se n=m+1 ¢,,» because by hypothesis we may, 8 We have here replaced each factor in the denominators by the least factor, i. e. by 2. * German: Vergleichskriterien. (Tr.) ? Gauss used this criterion in 1812 (v. Werke III, p. 140). It was not, however, formulated explicitly, nor was the following test of the 2° kind, be- fore Cauchy, Analyse algébrique (Paris 1821). v3. 114 Chapter III. Series of positive terms. for every in >m, write q,=y,¢,, with y, <1. In case b) the di ® vergence results similarly from that of 3 d , because here we may n=m+1 : wilte nw =0,4d, with 4. >15% nn? Comparison test of the 2" Lind. Let 2c, and 2d, again denote respectively a convergent and a divergent series of positive terms. If the terms of a given series Za, of positive terms satisfy, for every mn => a certain m, a) the conditions LE | < Cnt1 == ’ a, Cn then the series 2a, is also convergent. If, however, for every n> a certain m, we have b) constantly Dnt, Init By a, : then 2a, must also diverge. Proof. In case a), we have for every n > m @n +1 = Za Cn+1 = Cn . ay . . . The sequence of the ratio y,= — is, from a certain point on- n wards, monotone descending, and consgquently, since all its terms are positive, it is necessarily bounded. Theorem ¢0, 2 now establishes the a, a. convergence. In case b) we have, analogously, me +=, so that the n+1 n . a . . ratios J, = = increase monotonely from a point onwards. But as they n are constantly positive, they then have a positive lower bound. Theo- rem 70, 3 now proves the divergence. These comparison tests or criteria can of course only be useful to us if we are already acquainted with a large number of convergent and divergent series with positive terms. We shall therefore have to lay in as large a stock as possible, so to speak, of series whose con- vergence or divergence is known. For this purpose the following examples may form a nucleus: 8 Or else — almost more concisely —: In case a) every bound above of the partial sums of X¢, is also one for the partial sums of Xa,; and in case b), the partial sums of Xa, must ultimately exceed every bound, since those of Xd, do so. a me § 12. The first principal criterion and the two comparison tests. 115 1 ; 1 Z 1. Y= was seen to be divergent, D>' — convergent. By the first com- 74. n=1 7 n=1 7 parison test, the so-called harmonic series n=1 n? is therefore certainly divergent for « < 1, convergent for « 2. (Itis, however, only known in the case a = even integer how its sum may be related to num- bers occurring in other connections; for instance we shall see later on that for « =4 the sum is i 90° 2. By the preceding, the convergence or divergence of 3} 1, only remains n questionable in case 1 <1: To obtain a bound above for any partial sum s, of the series, choose k so large that 2% > xn. Then 1-1 1 1 pe =1 ee ’ i : wwe (got) (ger 0) + (rt Here we group in one parenthesis those terms whose indices run from a power of 2 (inclusive) to the next power of 2 (exclusive). Replace, in each pair of parentheses, every separate term by the first; this involves an increase of value and we have therefore Ak—1 S14 yg Pha LAE ;=9, — a positive number certainly <1, since ¢« >1, — then we have 1— 9k 1 = 2 .he Bign ra BEL at + +3 T5735 and since this holds for every =, the partial sums of our series are bounded, and the series itself is convergent, q. e. d. : ; 1 ; All harmonic series > = for « <1 ave divergent; and for a > 1, conver- n gent. In.these, with the geometric series, we have already quite a useful stock of comparison series. ~ 8. Series of the type Sel n=1 (an + b)* where a and b are given positive numbers, also diverge for ¢« <1, converge for «>1. For since n® 1. v2 1 1, —— l= ve have pote is @ntt)” a4 a (an +b) n n and 70, 4 proves the truth of our statement. 75. 116 Chapter III. Series of positive terms, Accordingly the series 1 1 z 1 14+4—4—+4...= rai ar - 2p Cn +1 in particular, are convergent for «> 1, divergent for « <1. @® 4. If > ¢, is a convergent series with positive terms, and we deduce fiom n=0 it a new series X¢,/ by omitting any of its terms and denoting those which remain by ¢/, ¢,’, ..., then the resulting “sub-series” X¢,’ is also convergent For every number which is a bound above for the partial sums of 3c, is then also a bound above for those of the new series. : 1 ; In accordance with this, the series oT , where p runs through all prime p integral values, i. e. the series 1 1 1 1 1 Fg teu is certainly convergent for ¢ >1. (On the other hand, of course, we cannot conclude without further examination that it diverges for « <1!) 5. Since Xa” is already recognised as convergent for 0 1 occurs at least G times while 0 0, and a +1 n then the series 2a, is convergent. If however, from some place onwards, a 41 21, Tn then the series 2 a, is divergent. (Cauchy’s ratio test?) Remarks and Examples. 76. no 1. In both these theorems, it is essential for convergence that Va, and Sty ay does not at all suffice for convergence that we should have respectively should be ultimately less than a fixed proper fraction a. It n_. a Veo, <1, of aiid g n : : . : 1 for every nm. An example presents itself at once in the harmonic series Sg for which we certainly always have n 2 a 1 1 1 = and also alm =r 1. For suppose Va, — a < 1, for instance; then & = 1 =>0 and m may be determined so that, for every » > m, we have n__ 1+ V aa 1, then ¢=221 > 0, and mw’ may be so determined that, for every n> m/, we have Var >a—v=11%a. . And since this value a is > 1, theorem 1 proves the divergence. — The proof in the case of the ratio is quite analogous. If « =1, these two theorems prove nothing. 3. The reasoning just applied in 2. is obviously also legitimate when is>1, lim Van or lim fun is <1, in the one case, and lim rw or lim et n in the other. Ito oe of these upper or lower limits is = 1, or the upper limit > 1, the lower <1, then we can infer nothing as to the convergence or diver- gence of 3 a,. The supplementary note tec 75, 1 however shows that, in the el root test, it is sufficient for divergence that lim Va, >1.11 4, The remarks just made in 2. and 3. are so obvious that, in similar cases in future, we shall not specially mention them. 5. The root and ratio tests are by far the most important tests used in practice. For most of the series which occur in applications, the question of convergence or divergence can be solved by their means. We append a few examples, in which z, for the present, represents a positive number. a) Sn*a™ (a arbitrary). Here we have Tuts _ ihe % ori, ay n A | 1 : ioe a gy = =1 desl and is permanently positive (v. 88, 8). The series is therefore — and this without reference to the value of « — convergent if x <1, divergent if x>1. For z=1 our two tests are inconclusive; however we then get the harmonic series, with which we are already acquainted. AED a Dna : b) = ( » IE = oy 1) ( 5 E (p an integer > 1). Here we have by Got DL)... 01D pl nlp t] Tiim a, m+p)(n—14+p)...(n+1)-p! n+ 1 : » 11 Thereby the criterion obtains a disjunctive form. 2a, is convergent A or divergent according as lim}/a, is <<1 or >1. (Further details in §§ 36 and 42.) § 13. The root test and the ratio test. 119 Hence this series too is convergent for x <1, divergent for a> 1, whatever : : . : a be the value of p. For z=1 it obviously diverges, since then a> 1 for n every n. In the case of convergence we shall later on find for its sum the 1 \p+1 value . c) > = =1tot? a cee —~ — ! Here we have for every x > 0 Th 20 (<1) an n + the series is therefore convergent for every z=0. For the sum we shall later on find the value e?. d 7 ZR £ > Ly x ) 2 oF is convergent for 220, as |/ 2 =_——0. Bo is convergent, as again Va, — 0.12 0) Fn Ta 1 f) Sr convergent, because ay <5 n! 1.2 ...0 2 iE = <= > = convergent, because a, SE TTS for every w= 2; 1 1 —————— divergent, because a, > yr (n+ 1) nl ra convergent, because a, 0, is divergent, since by 38, 4 from some # onwards (log a 5 n. h) 70 is divergent, because }/ n — 1. n_._- ny n . 1 i) Pee (log 2) 03% writing the generic term in the form is convergent, as we may at once recognize by 1 ploglogn’ 12 In this series, summation may only begin with #» =2, since log 1 =0. Such and similar obvious restrictions we shall in future not always expressly mention; it suffices, for the question of convergence or divergence, that the indicated terms of the series, from some place onwards, have determinate values. — In all that follows, unless the contrary is expressly stated, the sign “log” will always stand for the natural logarithm, i.e. that to the base ¢ (46a). ~ . = “i. 120 Chapter III. Series of positive terms. On the other hand 1 (log n)loglogn = is divergent, because by 38,4 and Ex. 13, (loglog#)? —. n § 14. Series of positive, monotone decreasing terms. Before passing from these quite elementary considerations, we will mention a particularly simple class of series of positive terms, namely those series whose terms q,, at least from some place onwards, form a monotone sequence. To this class belong nearly all the series given as examples above and also the majority of those which occur in applications. For such series we have the following: @ Cauchy’s theorem of convergence. If > a is a series whose terms form a positive monotone decreasing sequence (a,), them it con- verges and diverges with Se =0,+20, + da, 1-84, + -- Preliminary remark. What is particularly remarkable in this theorem is that it shows that a small proportion of all the terms of the series suffices to determine the convergence or divergence of the whole Series: For this reason it is also called the condensation Hoos, It shows that the harmonic series id for instance, is certainly diver- gent, for it converges and diverges with the series 2% opel Ly. : : 1 which is unmistakably divergent. And speaking generally, the series ~= As n inferred to converge and diverge with the series k 3 gy=2 (7): but this is a geometric series and therefore converges or diverges according aseo>l oral, These examples also show us that the convergence or divergence of & 2 8%. is often more easily ascertained than that of the series X a, itself; it is just in this that the value of the theorem lies. Proof. We denote the partial sums of the given series by s,_, those of the new series by #,. Then we have (cf. 74, 2) 13 Analyse algébrique, p. 1385. § 14. Series of positive, monotone decreasing terms. 121 2) for wn < 2° Su < y+ (ay Fag) + oo (ag EF gay) 2F Sp > ay + ay + (ag + a) + 0 A (Agios yy Foor + ag) >a, +a,+ 2a, + + 4-2F-1ay =1¢, ie 2s, >t. Inequality a) shows that the sequence (s,) is bounded if the sequence (#,) is bounded; inequality b), conversely, that if (s ) is bounded, so is (¢,). The two sequences are therefore either both bounded or both un- bounded, and therefore the two series under consideration either both converge or both diverge, q.e. d. Before given further examples illustrating this theorem, we may extend it somewhat!*; for it is immediately evident that the number 2 plays no essential part in the theorem. In fact we have, more generally, the Theorem. If Xa, is again a series whose terms form a positive 78. monotone decreasing sequence (a), and if (gy, gy ---) 1S any monotone increasing sequence of integers, then the two series > Lo and = (8rs1 go a, n=0 k=0 k are either both convergent or both divergent, provided g,, for every k > 0, fulfils the conditions G7 020 ad 5, ~5 SMls —g_) in the second of which M stands for a positive constant®. Proof. Exactly as before we have a) for n < g,, — denoting by 4 the sum of the terms possibly preceding a, (or otherwise 0), — S, < sg, <4 lopli cha) hi +0. ses ols ay, 1) Sd dg, =~ ga, + ore dln — 0) ag, 5, < A441, 1 Schiomilch, O.: Zeitschr. f. Math. u. Phys., Vol. 18, p. 425. 1873. 1» The second condition signifies that the gaps in the sequence (gj), re- latively to the sequence of all positive integers, must not increase at too great a rate. Fi v9. 122 : Chapter III. Series of positive terms. -* b) for n> g, 5, 28, > (Gey thay) rik lay yt tay) Sloot lg 5a) ag, > Ms, > ~2)%,+ + 6 — &) aq, Ms. >t — 1. And from the two inequalities the statements in question follow in the same way as before. Remarks. 1. It suffices of course that the conditions in either theorem be fulfilled from and after a definite place in the series. Therefore we may, in the extended theorem, suppose, as a particular case, = L=8 di 0 or ={gf] where g is any real number >1 and [g¥] the largest integer not greater than g¥. We also satisfy the requirements of this theorem by taking G=k= = hi 0. 0 With gr =k? we obtain, for instance, the theorem that the series >} a,, — if 8k ny n=0 (@,) is a positive monotone decreasing sequence, — converges and diverges with Rr] a.=0a,+3a,+5a,+ Ta, + --- We may also replace this last series, according to 70, 4, by the series Shap=a,+2a,+8ay+--- Z 3 n=2 those of the harmonic series; for according to our theorem, this series con- verges and diverges with is divergent, — although its terms are materially less than n log n oF gy = 2%. Jog (2%) mp (log 2)-k and is therefore, by 70, 2, like the harmonic series, divergent. The divergence of this series and of those considered in the next examples was first discovered by N. H. Abel'® (v. (Euvres II, p. 200). 0 1 - 3. = Sinha tony is also still divergent, although its terms are again considerably less than those of the Abel's series just considered. For by Cauchy's theorem it converges and diverges with 2] of 0 1 i= 2% log ok. log (log 2%) =2 klog 2-log (klog 2) 2 18 Niels Henrik Abel, born Aug. 5%, 1802, at Findoe near Stavanger (Nor- way), died April 6%, 1829, at the Froland ironworks, near Arendal. § 14. Series of positive, monotone decreasing terms. 123 discussed and this, since log 2 << 1, has larger terms than Abel's series >] % a 5 above, and must therefore diverge. 4. Thus we may continue as long as we please. To abbreviate, let us denote by log, x the 7P* repeated or iterated logarithm of a positive number z, so that logyz=2, log, x=log>, log, ax=log(loga),... log, z = log (log, 2). We may also take log_, x to denote the value e”. These iterated logarithms only have a meaning if x is sufficiently large; thus log z only for 2 >0, log, = only for #>'1, log, = only for x >¢, and so on; and we shall only place them in the denominators of the terms of our series if they are positive, i. e. log z only for x > 1, log, x only for x >e, log, z only for x > e?, and so on. If therefore we wish to consider the series 1 5 ; Se integer > 1 2 Tog nlog nog, 7 @ ger 21), then the summation must only begin with a suitably large index, — whose exact value, however, (by 70, 1), does not matter. Since the logarithms increase monotonely with #, and the terms therefore decrease monotonely, the series, by Cauchy’s theorem, converges and diverges with 1 ST log, 2% and this, since 2 < e, must certainly diverge, if 1 = Flog. og F diverges. Since the divergence of the latter series was proved for p=1 (and p=2), it follows by Mathematical Induction (2, V) that it diverges for every p= 1. 5. The series atove considered, however, become convergent if we raise the last factor in the denominator to a power >> 1. That vl converges for ne «>> 1, we already know. If we assume proved for a particular (integer) p => 1, that the series 1 17 1 *) St vor log, b-(log, , RY* eal is convergent, it follows just as before that the series 1 n #n-logn...log, ,n-(log,n)* {o>1) is also convergent. For this, by the extended Cauchy's theorem 78, converges and diverges with the series — we choose g; = gts gh+1__ gk Z 310g 2%... (log, 3%)" . -— al 17 For p= 1, this reduces to the series LAE k 124 Chapter III. Series of positive terms. As 3>e, this series has its terms less than those of the series (*) (assumed _ convergent), if the terms of the latter are multiplied by 2 (which by 70, 2 80. leaves the convergence undisturbed). The series brought forward in the two last examples will later on render us most valuable services as comparison series. We will prove one more remarkable theorem on series of positive monotone decreasing terms, although it anticipates to a certain extent the general considerations on convergence of the following chapter (v. 82, Theorem 1). Theorem. If the series 2 a, of ovis monotone decreasing terms is to converge, then we must hte not only a,— 0, but na,—0.% Proof. By hypothesis, the sequence of partial sums a, +a, + --- -+ a, =s, is convergent. Having chosen ¢ > 0, we can therefore so choose m that for every » > m and every 1 >1 we have Sti Seb | ov+4 » 9? aiih nt Fau 2m, then, taking » = [5 n], the largest integer not greater than lz, we have » > m and therefore Guy + intr Ta, < : a fortiori, therefore, n—va, << - and At wlt 0 a we, Therefore na,— 0, q. e. d. Remark. We must expressly emphasize the fact that the condition na,— 0 is only a necessary, not a sufficient one for the convergence of our present type of series, i. e. if na, does not tend to 0, then the series in question is certainly divergent!®, while na, —> 0 does not necessarily imply anything as to the possible convergence of the series. In point of fact, the Abel's series Sol diverges, although it has monotone decreasing terms and n log n 1 — 0. log n na, = 18 Olivier, L.: Journ. f. d. reine u. angew. Math., Vol. 2, p. 34. 1827. 5 : 1 : ii 19 Accordingly, the harmonic series a for instance, must diverge 1 because it has monotone decreasing terms, but Bias does not tend to 0. Exercises on Chapter IIL 123 Exercises on Chapter III. 34. Investigate the behaviour (convergence or divergence) of a series > a,, for which a,, from some index onwards, has the following values: 1 nt n+ yn (nl)? 27.31 37.n! [ie SEL nl’ nei—mn' 2n)l’ nr’ n? nw n wo "w.. 1 Be 8 re (em), Fe-1-3) wim, (sive), va +1— Vr 1 1 Vr, alogn gloglogn % (log log w) '*E™' (logy n)'%¢"’ ; (gy BEG) 85. If 3d, diverges, so also does >) What is the behaviour of dn itd, Sr d a is ad, 0. A oie a TTL (a ) 86. Under the same assumption that Xd, diverges and d, > 0, what is the behaviour of the series Sri 87. Suppose p, —-+ oo. What is the behaviour of the series 1 1 1 SE! = PRCT 2 ogtogn’ 38. Suppose p, — 0, but with 0 < Tim (pppy — pn) < +0. What must be the upper and lower limits of the sequence (go,) so that x 1 Pa Cn converge or so that it diverge? 89. For every n> 1, n rr. 7 1 wld do dep bist gpiry lh 40. The sequence of numbers Wy = [54 +e +1 10pm] is monotone descending, 41. If Ya, has positive terms and is convergent, then X'ya, a, , is also convergent. Show by an example that the converse of this theorem is not true in general, and prove that it does nevertheless hold when (a,) is monotone. / 42. If Ya, converges, then yin also converges, and also indeed the series Ts for every 6 > 0. ww Si. 126 Chapter IV. Series of arbitrary terms. 43. Every positive real number #, is, in one and only one way, ex- pressible in the form mat where a, is a non-negative integer with a, a, is that, having chosen any €¢ > 0, we can assign a number ny = n, (&) such that for every n> mn, and every k > 1, we have | Sntru— Sn | < &, E that is to say, in the present case, that | Any1+ pyr : + ang | < &. § 15. The second principal criterion and the algebra of convergent series. 127 Starting with the second form of the main criterion, we also ob- tain for the present fundamental theorem the following oSecond form. The series 3 a, converges if, and only if, given 81a. a perfectly arbitrary sequence (k,) of positive integers, — the sequence of numbers ) I.=— (41 + Up 42 + ane + Antr,) invariably proves to be a null sequence. And as before we can extend this somewhat to the OThird form. The series > a, converges if, and only if, given S1B. two perfectly arbitrary sequences (v,) and (k,) of positive integers, of which the first, at least, tends to - co, — the sequence of numbers In=@, +14+d, 21+ +--+ Wii) invariably proves to be a null sequence. Remarks. 1. A series represents essentially a new symbolic expression for se- quences of numbers, and in particular, as we remarked, not only every series represents a sequence, but every sequence is also expressible as a series; all remarks and examples given on p. 82 seq. have their parallels here. 2. The contents of the fundamental theorem may be formulated as follows: Given & > 0, every portion of the series, however long, provided only its initial index be sufficiently large, must have a sum whose absolute value is 0, we must be able to assign an index m so that for » > m the addition of an arbitrary number of further terms to s, can never alter this partial sum by more than e. 3. Our present theorems and remarks of course also hold for series of positive terms. This the reader should verify in each separate case. A finite part of the series, such as Ayirt + Gyro + r+ Ayia we may for brevity call a portion of the series, denoting it by 7, if it begins immediately after the »* term. When required, we may further ex- plicitly indicate the number of terms in the portion by denoting this by T,,;,. lf we are considering an arbitrary sequence of such portions whose initial index — -- co, we shall refer to it for short as a “se- quence of portions” of the given series. The second and third form of the fundamental theorem may then also be expressed thus: oq form. The series 2 a, converges if, and only if, every Sle. “sequence of portions” of the series is a null sequence. 1 It is substantially in this form that N. H. Abel establishes the criterion in his fundamental memoir on the Binomial series (Journ f. die reine u. angew. Math., Vol. 1, p. 311. 1826). » \ 128 : Chapter IV. Series of arbitrary terms, Remarks and examples. 1. Za, is thus divergent if, and only if at least one sequence of portions can be assigned which is not a null sequence. For the harmonic series 1 ; > —, for instance, we have n 1 1 +oTH The sequence (7,) is therefore certainly not a null sequence, and therefore 1 1:1 er hgn > Grp Ean L=Ten® oy AY te is divergent. 2. For 3 we have 1 1 Lebo=gzt sim eda Fa, od yor) CFD +2 CFi-D GD AT 1 1 1 1 1 1 1 =r) bam) tt rr) my 1 : therefore T, << —, so that 7,— 0, when »— +00. The series therefore v : converges. 3. For the sequence wor v1.2. 1..1 pes pnitt we have Me 1 1 1 3h Tom Toy = tr [hy lpr lpg EI : Whether % is even or odd, the expression in brackets is certainly positive and 1 Sarl ing negative term, the sum of the two is in each case positive. If k is even all terms are exhausted in this manner, if 2 ¢s uneven a positive term remains, so that in either case the complete expression is seen to be positive. If, on the other hand, we write it in the form For if we take together, in pairs, each positive term and the follow- Sorelle) limi n+ 1 n+2 n+t3 n+4 n-t+d z all the terms are now exhausted when % is odd and a negative term remains over if k is even, so that in both cases only subtractions from 1 Zid occur, and thus the expression is n, and every k = 1, allow k to increase beyond all bounds and so obtain, for every n > n,,7, <¢. Thus we have the 0 OTheorem 2. The remainders rv, = > a, of a convergent series % v=n+1 Da, =s, — i.e the numbers n=0 ! (0 == 8), Por ys Tasers Vy ds dey = always form a null sequence. In 80, we saw further that if the terms of a convergent series Sa, (of positive terms) are monotone decreasing, then, over and above the theorem just proved, the condition na, —0 must hold. That this need no longer be the case in series of arbitrary terms is already shewn by the series given in 81 e, 3. We can, however, show that we must have : — i. e. that the terms of the sequence (na,) are small on the average. In fact we have the more general eel OTheorem 3. If >a, ts a convergent series of arbitrary terms n=0 and if (py, py» ++.) denotes an arbitrary monotone increasing se- quence of positive numbers tending to - co, then the ratio Py ay + Py a +p,a, (0.3 D, We 2 L. Kronecker, Comptes rendus de 1'Ac. de Paris, Vol. 103, p. 980. 1886. — Moreover, this condition is not only necessary, but also, in a quite deter= minate sense, sufficient (cf. Ex. 58a). 82. 130 Chapter IV. Series ot arbitrary terms. Proof. = By 44, 3, s, —»s implies PiSot BaP) 511 (Pa —Pn—1) Sher, 5 Ss: Since Bs — 0 and s,—s, we must therefore have 5, — (br — Po) So + (bs — £1) Ba + (Pa “Progin-y Lg. But this is precisely the relation we had to prove, as may be seen at once by reducing to the common denominator p and grouping in succession the terms which contain p,, p,,..., p, respectively? As regards any condition for convergence whatsoever, we have to repeat expressly that the stipulations made therein always concern — or only need concern — those terms of the series which follow on some determinate one, whose index may moreover be replaced by any larger index. In deciding whether a series is or is not con- vergent, the beginning of the series, — as itis usually put for brev- ity, — does not come into account. This we express more exactly in the following @® OTheorem 4. If we deduce, from a given series > a n=0 a new n? series > a,' by omitting a finite number of terms, prefixing a finite n=0 number of terms, or altering a finite number of terms (or doing all three things at once) and now designating afresh the terms of the series so produced by a), a, ...,* then either both series converge or both diverge. Proof. The hypotheses imply that a definite integer 9=0 exists such that from some place onwards, say for every # > m, we have Every portion of the one series is therefore also a portion of the other, provided only its initial index be > m -}- | ¢ | . The fun- damental theorem 81a immediately proves the correctness of our statement. 3 Instead of the positive p, we may (cf. 44, 3 and 5) take any se- quence (p,), for which, on the one hand, | p, | > + 00 and, on the other, a constant K is assignable for which | ?o jl lt tl ter | < Ela] for every n. 4 I. e. in short: “.. by making a finite number of alterations (27, 4) in the sequence (a,) of the terms of the series...” § 15. The second principal criterion and the algebra of convergent series. 131 Remark. It should be expressly noted that for series of arbitrary terms, compari- son tests of every kind become entirely powerless. In particular, of two series Sa, and Xa,’ whose terms are asymptotically equal (a, 2 a,’), the one may = quite well converge and the other diverge. Take for instance dy 1 . n and a, =a, + — ET oloow Finally we prove the following criterion of convergence, which appears almost unique in consequence of its particularly elementary character, and relates to the so-called alternating series, i. e. to series whose terms have alternately positive and negative signs: Theorem 5. [Leibnitz’s rule] An alternating series, for which the absolute values of the terms form a monotone null sequence, is tnvariably convergent. The proof proceeds on quite similar lines to that of 81e, 3. For if 3a is the given alternating series, then 4, has either the sign (— 1)", for every nu, or the sign (—1)*+1, for every n. If we write, therefore, | a, | = o,, we have = Tor w= [14 — Chie TCs 3 on + {~ le a) As the «'s are monotone decreasing, we may convince ourselves precisely as in the example referred to, that the value of the square bracket is always positive, but less than its first term e,,,. Thus } 2] =| Tl < Cuts which, since e¢, forms a null sequence by hypothesis, involves 7, — 0 and therefore convergence of 3 a,, by 81e. The algebra of convergent series. Already on p. 102 it has been emphasized that the term “sum”, to designate the limit of the sequence of partial sums of a series, is misleading in so far as it arouses a belief that an infinite series may be operated on by the same rules as an (actual) sum of a definite number of terms, e.g. of the form (a 4 b 4c +d), say. This is not the case, however, and the presumption is therefore fundamentally erroneous, although some of the rules in question do actually remain valid for infinite series. The principal laws in the algebra of (actual) sums are (according to 2, I and. III) the associative, distributive and commutative laws. The following theorems are intended to show how far these laws remain true for infinite series. 5 Letters to J. Hermann of 26. V1. 1705 and to John Bernoulli of 10.1. 1714. S3. 132 Chapter IV. Series of arbitrary terms. OTheorem 1. The associative law holds unrestrictedly for convergent infinite series; that is to say, ay + a, + a, + $ implies (2, + 2, 1 Tot +4.) + @+1t tet ta) t= if v,,9,,... denote any increasing sequence of different integers and the sum of the terms enclosed in each bracket is considered as ome term of a new series 4,4, +--+ 4, 4 where, therefore, for k=0,1, 2, Ay = Crt ht Forel, (ro = —1). Proof. The succession of partial sums S, of 34, is ob viously the sub-sequence s, , s,,,..., NOERER of the sequence of partial sums s, of Xa,. By 41, 4, S, therefore tends to the same limit as s,,. Remarks and examples. S11? 1.1 : 1. The convergence of > ~———=1——_4———... therefore im- = n 2:08 4 plies that of : 7 my REY ag pe iy al ay yan) = A EOmEmt atm and also, similarly, of 3-1-1) wl and all three series have the same sum. If we es this by s, the second : a 1 1 : series shows that in any case, s > y ine 7 — and the third, that 1 10 $l =gy=u. Thus 7 eld 10 i nS “p 2. Let us expressly remark that although, by Theorem 1, we may in- troduce brackets, we may not without consideration omit brackets occurring in a series, — as the following simple example shows: The series 0 4+0+0+--- is certainly convergent and has the sum 0. If we substitute everywhere (1 —1) for 0, we obtain the correct equality -ND+1-D+...=31-1)= But by omitting the brackets we obtain the divergent series 141005 0, \ § 15. The second principal criterion and the algebra of convergent series. 133 which therefore may not be put “=0”. For we should then by again group- ing the terms, though in a slightly different way, obtain 1=-0-D=0 1 srz=1-0-0-0%., which again converges and has the sum 1. We should therefore finally deduce that 0'=111° We proceed at once to complete Theorem 1 by the following 0 OTheorem 2. If the terms of a convergent infinite series > A, E=0 are themselves actual sums (say, as above, Ay = ay 41 +--+ a, ; k=0, 1,...;%,= = 1), then we “may” omit the brackets enclosing these if, and only if, the new series >a, thus obtained also converges. n=0 In fact in that case, by the preceding theorem, 2q,=24,, while in the case of divergence of 2'a,, this equality would become meaningless. A usually sufficient indication as. to whether the new series con- verges, is provided by the following OSupplementary theorem. The new series 2a, deduced from 2 A, in accordance with the preceding theorem is certainly convergent if the quantities 4) = Ay, 41 | = Ay +2 ir svete ay, | form a null sequence’. Proof. If z be given > 0, choose m, so large that, for every k > m,, we have. & Sh 5

m,, we have 4,’ < = If m is larger than both these numbers m, and m,, then we have, for every n>, , | [ § 8 <¢. 6 In former times — before the strict foundation of the algebra of in- finite series (v. Introduction) — mathematicians found themselves fairly at a loss when confronted with paradoxes such as this. And even though the better mathematicians instinctively avoided arguments such as the above, the lesser brains had all the more opportunity of indulging in the boldest speculations. — Thus e. g. Guido Grandi (according to R. Reiff, v. 69, 8) believed that in the above erroneous train of argument which turns 0 into 1, he had obtained a mathematical proof of the possibility of the creation of the world from nothing! ? As Ay — 0, this is of itself the case if the terms which constitute Aj have one and the same sign — in particular, therefore, if by omission of the brackets we obtain a series of positive terms. 134 Chapter IV. Series of arbitrary terms. For to each such # corresponds a perfectly definite number %, for which 1 m. In that case, however, S=Ss tt nt ta, i a J eh And since Sn = = (=, = Sie) Je (Sia = s) we then have, effectually, ls, — 8] <2, Lhe Ne=35, ged n=0 Pn 1 3 [2 1 > ( 1 1 BD) Ady = a st ad al nnd STIL LE eV i = (1+3 p/h) tor st 0 is convergent; for 4; is positive, and, for every £>1, is Peal Satay 0g 47-1 2; IG-Dr=1% Since similarly, for every 2 >1, : a 1 Smit iy (4x) is a null sequence. Therefore the series 1 pel re iE T +b — eee Ot 1 oT ls 1 is also convergent. — Its sum — call it S — is certainly Zeal, as the series in its first form had only positive terms. ©Theorem 3. Convergent series may be added term by term. More precisely, Dl, =8 and - Tb w= n=0 n=0 umply both Sle nt 0)=s+1 and also — without brackets! — Grr ath ta, =3-1-1, § 15. The second principal criterion and the algebra of convergent series. 135 L. Proof. If 5 and ¢ are the partial sums of the first two series, then (s -}-£) are those of the third. By 41,9, it therefore follows at once that (s, + #,)—s-¢ That the brackets may be omitted, in the series thereby proved convergent, follows from the supplementary theorem of Theorem 2, since (|a,|) and (|b,|) and therefore also (|a,|+ |b,|) are null sequences. ; OTheorem 4. Convergent series may in the same sense be sub- tracted term by term. The proof is identical. OTheorem 5. Convergent series may be multiplied by a constant, that is to say, from 2a,=s it follows, if c is an arbitrary number, that Sea) =¢s. Proof. ‘The partial sums of the new series are cs, if those of the old are s,. Theorem 41, 10 at once proves the statement. — This theorem, to some extent, provides the extension to infinite series of the distributive law. Remarks and Examples. 1. These simple theorems are all the more important, as they not only provide the convergence of the new series, but also set up a relation between its sum and that of the old series. They form therefore the foundation for actual calculation in terms of infinite series. os SET 2. The series te was convergent. Let s denote its sum. By n=1 theorem 1, the series us 1 1 ) = ( 1 1 1 1 ) a a a Surya are then also convergent with the sum s. Multiply the first by %, in accor- dance with Theorem §, — this giving & 1 1 s en jac =2 4p 2 — and add this term by term to the second; we obtain “(1 L =i) tre = or more precisely: we obtain the convergence of the series on the left hand side and the value of its sum, — the latter expressed in terms of the sum of the series from which we started. The convergence was also proved directly in - connection with theorem 2; the present considerations have led however appre- ciably further, since they afford a definite statement as to the sum of the series. Before we examine the validity of the commutative and distributive laws and investigate, in relation to the latter, the possibility of forming the product of two series, we still require an important preliminary. 86. 136 Chapter IV. Series of arbitrary terms. § 16. Absolute convergence. Derangement of series. The series 1 — 341 — 24... proved (81¢, 3) to be_convergent. But if we replace each term by its absolute value, the series becomes the divergent harmonic series 1+ 24 £4 ---. In all that follows, it will usually make a very material difference whether a convergent series Xa, remains convergent or becomes divergent, when all its terms are replaced by their absolute values. Here we have, to begin with, the OTheorem. A series Xa, is certainly convergent if the series (of ‘positive terms) Za, | convergesS. And if Za, =s, Z|a,|=S then BE Proof. Since la tt, nti, S00) Led the left hand side is here certainly < ¢ if the right hand side is, whence by the fundamental theorem 81 our first statement at once follows. Since further Is. | m, satisfy the condition ‘nti ry, he a la,| Lc, or the condition Es : _ an To. bn then Xa, is (absolutely) convergent.*® Proof. By the 1% and 27d comparison tests, 72 and 73, respec- tively, X|a,| is in either case convergent, and so therefore, by 85, is Ta. lt In consequence of this simple theorem the complete store of con- vergence tests relating to series of positive terms becomes available for series of arbitrary terms. We infer at once from it the following OTheorem 2. If Za, is an absolutely convergent series and if the factors a, form a bounded sequence, them the series 2a a n is also (absolutely) convergent. Proof. Since (|e,|) is a bounded sequence simultaneously with (ee), it follows from 70, 2 that Xe, | -|a,|=2 |e, a,| is convergent ~ simultaneously with Xa, |. Examples. 1. If 3¢, is any convergent series of positive terms and if the e, 's are bounded, then X ,c, is also convergent, for then X¢, is also absolutely con- vergent. We may thus, for instance, instead of joining the terms c¢y,c¢,, cq... with the invariable sign --, replace this by quite arbitrary + and — signs, — in every case we get a convergent series; for the factors 4 1 certainly form a bounded sequence. Thus for instance the series Se, snlml,,, I= Heer, are all convergent, where [2], as usual, stands for the largest integer not greater than z. 10 In the second condition, it is tacitly assumed that, for every =» > m, a, 3 0and c, =: 0. 11 The corresponding criteria of divergence, dnt1 n 41 yn are of course abolished, since the divergence of X'|a, |, not necessarily of Xa,, is all that follows. Cf. Footnote 8. la, 2d, and » = n 8S. 138 : Chapter IV. Series of arbitrary terms. 2. If Ya, is absolutely convergent, then the series obtained from it by an arbitrary alteration in the signs of its terms, is invariably an absolutely convergent series. We shall now — returning thereby to the questions put aside at the end of last section (§ 15), — show that for absolutely convergent series the fundamental laws of the algebra of (actual) sums are in all essen- tials maintained, but that for non-absolutely convient series this is no longer the case. Thus the commutative law “a-b=>b- a” does mot in general hold for infinite series. The meaning of this statement is as follows: If (»,,%,,%,...) is any rearrangement (27, 3) of the sequence (0,1, ..): then the series @D 0 Sa =a, de wihg' sa forn=012..) n=0 a=0 # geo] will be said, for brevity, to result from the given series a by n=0 rearrangement or derangement of the latter. The value of (actual) sums of a definite number of terms remains unaltered, however the terms may be rearranged (permuted). For infinite series this is no longer the case'®. This is shown already by the two series considered as examples in 81¢, 3 and 83 Theorems 1 and 2, namely 1—t+2—-24+—... and 1+3—3 +: +2 —14+F —-- which are evidently rearrangements of one another, but have different sums. The sum of the first was in fact s << 13, while that of the second was si > +2 iL; ‘and indeed the considerations of 84, 2 showed more precisely that s' = 3s. This circumstance of course enforces the greatest care in working with infinite series, since we must — to put it shortly — take account of the order of the terms!3 It is therefore all the more valuable to know in which cases we may not need to be so careful, and for this we have the OTheorem 1. For absolutely convergent series, the commutative law holds unrestrictedly* Proof. a) We first prove the theorem for series of positive terms. Let X¢,, having the partial sums s, and sum s, be such a series, and 12 This was first remarked by Cauchy (Résumés analytiques, Turin 1833). 13 As Ja, merely represents the sequence (s,), and a rearrangement of Sa, produces a series Xa,’ with entirely different partial sums Sp’, — these not merely forming a rearrangement of (s,), but representing entirely diffe- vent numbers!! — it seems a priori most improbable that such a derangement will be without effect on the behaviour of the series. 14 Lejeune-Divichlet, G.: Abh. Akad. Berlin 1837, p. 48 (Werke I, p. 319). Here we also find the example given in the text, of the alteration in the sum of the series by derangement. = § 16. Absolute convergence. Derangement of series. 139 let S¢c'=2 c, » of partial sums s, be an arbitrary rearrangement of itt Then ; Sx < Sy if N is taken larger than all the numbers »,, »,,v,,...,,} for on the right hand side appear, amongst others, all the terms which form the sum on the left. Since sy 0, first choose m, in accordance with 81, so large, that for every £ > 1 12s) Tanta, le and now choose #, so large that the numbers v,, ¥,, ¥,, +.., ¥p, COm- prise at. least all the numbers 0, 1, 2, ..., m5. Then the terms Gy Gys Uns » 245 4, Svidenily cancel in the difference s,’ — s,, for every n > ny, and only terms of index > m remain, — that is only (a finite number of the) terms a, +1° Bp sas ++ Since, however, the sum of the absolute values of any number of these terms is always < g, we have for every n> n,, |) —s, l= ‘and therefore (s,’ —s,) is a null sequence. But this implies that s)=s,+}(s,/ —s,) has the same limit as ¢,, i.e, Ja, and Za’ have the same sum, q. e. d.% 15 That such a number #», exists, follows from the very definition of derangement. 16 Alternative proof. Since Xa, and X|a,| converge, the series a, + an Ap | — ap iN bali and 7 balm also converge, by 83, 3, 4 and 5; let P and N be their sums. These are series of positive terms. (What do they and their sums represent?) Therefore by a), they admit of being arbitrarily rearranged, so that in particular we have ’ ’ pig plore Lid =p and Slo L eX, If — as we may, by 83, 4, — we subtract term by term the second of these series from the first, we obtain: Xa,’ =P — N, while the unaccented series give by subtraction: Xa, = P— N. Therefore 2 a,’ converges and has the same sum as Xa,, q.e. d. S9. 140 Chapter IV. Series of arbitrary terms. This property of absolutely convergent series is so essential that it deserves a special designation: 8 ODefinition. A convergent infinite series which obeys the commu- tation law without any restriction, — 1. e. remains convergent, with unaltered sum, under every rearrangement, — shall be called unconditio- nally convergent. A convergent series, on the other hand, whose be- haviour as to convergence can be altered by rearrangement, for which therefore the order of the terms must be taken into account, shall be called conditionally convergent. The theorem proved just above can now be expressed as follows: “Every absolutely convergent series is unconditionally convergent.” — The converse of this theorem also holds, namely OTheorem 2. Every mon-absolutely convergent series is only con- ditionally convergent’. In other words, the validity of the equality in the case of a non-absolutely convergent series Xa, depends essen- tially on the order of the terms of the series on the left, and may therefore, by a suitable rearrangement, be disturbed. Proof. It obviously suffices to prove that, by a suitable rearrange- ment, we can deduce from 2a, a divergent series 2g’. This we , may do as follows: The terms of the series 2g, which are > 0, we denote, in the order in which they occur in 2a , by p,, p,, Pivsecl those which are << 0 we denote similarly by —g¢,, —¢,, — Gyssve Then 2p, and 2g, are series of positive terms. Of these, one at least must diverge. For if both were convergent, with sums P and Q say, then we should obviously have, for each #, latte tick ln I< PLO, hence 2'a,, would, by 70, be absolutely convergent, in contradiction with our assumption, If for instance Xp diverges, then we consider a series of the form By me Pte ti = 0b Pst Pad oo i ly Bag ee, in which, therefore, we have alternately a group of positive terms followed by a single negative term. This series is clearly a rearrange- ment of the given series Xa, and will, as such, be denoted by Z¢q ". Now since the series 2'p was assumed to diverge, and its partial sums are therefore unbounded, we can, in the above, first choose m, so large that p+ 5, 1 +++ + Py, > 1 +g, , then m, > m, so large that 17 Cf. Fundamental theorem of § 44. 18 Jt is not difficult to see that actually both the series Xp, and Xg, must diverge (cf. § 44); but this is for the moment superfluous. § 16. Absolute convergence. Derangement of series. 141 htt ttn tr Tm 2 0TG and, generally, m, > m,_, so large that Put lt: tho? th Th ta,, (»=3,4,...). But Za is then clearly divergent; for each of those _ partial sums of this series whose last term is a negative term — g, of Za, is by the above >» (r=1,32,...) And since » may stand - for every positive integer, the partial sums of Yq ' are certainly not bounded, and Xa itself is divergent, q.e. d.?? If Yq, is divergent, we need only interchange 2p, and 2g, suitably in the above to reach the same conclusion. Example. 3 tr =-—1 tiz Pel is -.. was seen to be non-absolutely con- vergent. Since (cf. 46, 3), for 1=1, 2, ..., : P11 1 A alaiey har we have, for v=1, 2, ... 1 98% toy. ghptpde tna, Tye If therefore we apply to the series Soir the procedure described above, we need only put m, = 8», to deduce from it by rearrangement the divergent series I. 1, 1 1 1 7 ghytgt molt totm-gphe For the partial sums of this series -terminating with the »' negative term is greater than 2» minus » proper fractions, — i. e. certainly >». Theorem 88,1 on the derangement of absolutely convergent series may still be considerably extended. For the purpose, we first prove the following simple OTheorem 3. If 2a, is absolutely convergent, them every “sub- series” 2a, — for which the indices A, denote, therefore, amy mono- tone increasing sequence of different positive integers, — is again convergent and in fact again absolutely convergent. Proof. By 74,4, X|a,| converges with Xa). By 83, the statement at once follows. We may now extend the rearrangement theorem 88,1 in the following manner. We begin by picking out a first sub-series 3a; of the given absolutely convergent series Xa, and arranging this first sub-series in any order, denote it by 00 of gO LL 19 Ya,’ clearly diverges to + 00. we 142 Chapter IV. Series of arbitrary terms. let 2 be the sum of this series, certainly existing, by the preceding theorem, and independent of the chosen arrangement by 88, 12°. We may also allow this and the following sub-series to consist of only a finite number of terms, — i. e. not to be an infinite series at all. From the remaining terms — as far as is possible — we again pick out a (finite or infinite) sub-series, and denote it, arranged in any order, by 2 + a,® +4 a,@ +... 4a, 4... its sum by 2%; from the remaining terms we again pick out a sub- series, and so on. In this manner, we obtain, in general, an infinite series of finite or (absolutely) convergent infinite series: 0 0 0 2,® + a, © 1a Ot +2, @ is = SO a, +a bath La dott 4 Slob. | + a,® re = If the process was such as to give each term of the series Ja a place in one (and only one) of these sub-series, then the series 2© +4 2 =L 2® +... or, that is to say, the series (Z0,9) +(Ze,®) + (Za,®) + +s may in a further extended sense be called a rearrangement of the given series?!. For this again we have, corresponding to theorem 88, 1: OTheorem 4. An absolutely convergent series “may” also in the extended sense be rearranged. More precisely: The series 29 is again (absolutely) convergent, and its sum is equal to that of Xa. Proof. 1. By 85, we have 129 < | 2, @| + |, + -- lad #0] < lag + || + hl t 129] 0 be given, first determine m so that, for every k > 1, we have |a, | + | pial + +], <é& and then choose #, so that in the first n, -}- 1 sub-series 2a, y=0,1,...,%, the terms Ag) Ags Cys «+ +1 4, of the given series certainly appear. If # > nu, and > m, then the series n (2 | == vias + 3) ba contains only terms -+ a, whose indices are > m. Hence, by the choice of m, the absolute value of this difference is < ¢, and tends therefore, with increasing =, to zero, so that Bn (f® 4g tooo 1) = dims, = 5 =24,, Q. cd. n—r® n—>0 k=0 in other words, the two series formed by the sums of the rows and by the sums of the columns, respectively, are both absolutely con- vergent and have the same sum. : The proof is extremely simple: Suppose all the terms in (A) arranged anyhow (in accordance with 53, 4) in a simple sequence, and denoted, as terms of this sequence, by 4, 2,,4,,... Then Xa is absolutely convergent. For every partial sum of X|a, |, for instance |g Ela) teeta |, must still be < 0, since by choosing % so large that the terms a, a,, By; ++ 5 4, 3 ocCur in the 2 first rows of (A), we certainly have Jal + aoe [4 SEO 29 4 2, i.e. <0. A different arrangement of the terms 2,2 in (A) as a simple sequence a, a,’,a,’,... would produce a series X'g, which would be a mere rearrangement of 2'¢ , and therefore again absolu- tely convergent, with the same sum. Let this invariable sum be de- noted by s. Now both Xz® and also s™ are rearrangements of Sa =s, in the extended sense of theorem 4, just proved. Therefore these two series are both absolutely convergent and have the same sum s, q. e. d. This rearrangement theorem may be expressed in somewhat more general form as follows: OSupplementary theorem. If M is a countable set of numbers and there exists a constant K such that the sum of the absolute values of any finite number of the elements of M remains invariably < K, 22 Here the letter s is intended as a reference to the columns of (A). § 16. Absolute convergence. Dcrangement of series. 115 then we can assert the absolute convergence — with the invariable sum s — of every series 2 A, whose terms A, represent sums of a finite or infinite number of elements of M (provided each element of M occurs in one and only one of the terms A,). And this remains true if we allow a repetition of the elements of M, provided each ele- ment occurs exactly the same number i times in all the A,’s taken together, as in M itself®®. Examples of these important theorems will occur at several crucial points in what follows. Here we may give one or two obvious applications: 1. Let Ya, =s be an absolutely convergent series and put a+2a,+4a,+---+2%a, _, gn+1 Aan =0,1,2,) Then we also have Xa,’ =s. The proof results immediately, by the previous rearrangement theorem, from the consideration of the array a= 3 0 Toe fe LL + 2+ 2 a= GL a= 0+ 0 +4244 24. »isie Sec Sethe tel ete. Side e lle tere 1 il 1 0 REE SR LLL imilarly, from TD Ieee > (v ), and the array 0 les, xr oe st G=0 +25% Leg Lee m= 0 + 0 +377 hr 0 ®Eele eee we deduce the equality, valid for any absolutely convergent series Xa,: < 2 Bln ay + 2a, +8 a, 2] Lat Tht tia ht 3. The preceding rearrangement theorem evidently holds whenever every a,® is 220 and at least one of the two series Zz® and Xs converges; it holds further whenever it is possible to construct a second array (A’) similar to (A), whose terms are positive and > the absolute values of the corresponding terms in (A), and such that, in (A’), either the sums of the rows or the sums of the columns form convergent series, 2% An infinite number of repetitions of a term different from zero is ex- cluded from the outset, since otherwise the constant K of the theorem would certainly not exist. And the number 0 can produce no disturbance. 146 Chapter IV. Series of arbitrary terms. 91. 8 11. Multiplication of infinite series. We finally enquire to what extent the distributive law “a (b +c) = ab -} ac” holds for infinite series. That a convergent infinite series 2a, may be multiplied term by term by a constant, we have already seen in 83, 5. In the simplest form (Za) =21(ca) the distributive law is therefore valid for all convergent series. In the case of actual sums, it at once follows further, from the distributive law, that (a 4 b)(c+d)=ac-+ad-+bc+bd, and more generally, that (ata, +--+ a) +84 +5,)=2,0,1 a,b +--+ 4), or in short, that l m (2a) (20) = 3 wb, A=0 u=0 =0,.. 1 u=0,...,m where the notation on the right is intended to convey that the indices A and u assume, zndependently of one another, all the integral values from O to / and O to m respectively, and that all (I +1) (m + 1) such products a; b, are to be added, in any order we please. Does this result continue to hold for infinite series? If 2a, = and 2'b, =1¢ are two given convergent infinite series of sum sand ¢, is it possible to multiply out in the product in any similar way, and in what sense is this possible? More precisely: Let the products 1=0,1,2,... aby [foobhg be denoted, in any order we choose, by p,, p,, p,, ---2% is the series 2p, convergent, and if convergent, does it have the sum s-#2 — Here again absolutely convergent series behave like actual sums. In fact we have the OTheorem. If the series 2a, =3s and 2b, =1 are absolutely convergent, then the series 2p, also converges absolutely and has the sum s-t%. 2 We suppose, for this, that the products a, bi are written down exactly in the same way as a,® or ¢;(® for 53,4 and 90, to form a doubly infinite array (A). We can then suppose in particular the arrangement by diagonals or the arrangement by squares carried out for these products. 2 Cauchy: Analyse algébrique, p. 147. § 17. Multiplication of infinite series. 147 Proof. 1. Let » be a definite integer > 0 and let m be the largest of the indices 2 and u of the products a; 0b, which have been denoted by p,.9,;..., 9, Evidently tal +1 Foetal < ( Sha) ( 3101) i.e. «<< 0-7; if 0 and 7 denote the sums of the series X'|q;| and X|b,|. The partial sums of X'|p, | are therefore bounded and Xp, is ab- solutely convergent. 2. The absolute convergence of 2p having been proved, we need only determine its sum — call it S — for a special arrangement of the products a; b,, for instance the arrangement “by squares”. For this we have, however, obviously, ay by = Po> (+ a)(05 +0) =, +5, + 5: +B; and in general (y+ o lt ib BY = ov + Pip rairays an equality which, by 41, 10 and 4, becomes, when #— oo, $:l=2S which was the relation to be proved. Remarks and Examples. 1. As remarked, for the validity of the relation 3p, = s-¢ under the hypo- * theses made, it is perfectly indifferent in what manner the products a, b, are enumerated, that is to say arranged in order as a simple sequence (p,). The arrangement by diagonals is particularly important in applications, and leads, if the products in each diagonal are grouped together (83,1), to the following relation: @D oD 2 dn 2 bn = ay by + (a9 by +a, by) + (ag by + a, by +a, bg) ++ +» n= n= 0 2 Cn n=0 ll writing for brevity a,b,4-a,b,_, +a, b,_s+---+a,b,=c,. The validity of this relation is therefore secured when both series on the left converge ab- solutely. We are also led to this form or arrangement of the “product series”, sometimes called Cauchy’s product of the two given series26, by the conside- ration of products of rational integral functions and those of power series, which latter will be discussed in the following chapter: If in fact, we form the pro- duct of two rational integral functions (polynomials) dota, x+ta,xt+.--+ayxl and bytb zfby at... b,ax™ * Cauchy loc. cit. examines the product series in this special form only. 148 Chapter IV. Series of arbitrary terms. and arrange the result again in order of increasing powers of x, then the first terms are : ay by + (ay by +a, by) x + (ag by +a, by +a, bp) a? +--+, so that we have the numbers cy, ¢,, ¢,, +++, above introduced, appearing as coefficients. It is precisely due to this connection that Cauchy’s product of two series occurs particularly often. 5 2. Since Xa" is convergent for |x| <1, we have for such an x == San. Siar -Y@ +1”. 1—x n=0 n n 3. The series >, cf. 76, 5c and 85, is absolutely convergent for every real number xz. If therefore x, and x, are any two real numbers, we may form the product of Tn - rn Sa=Fi. a Phe n! ni according to Cauchy's rule. We get n no, BF n ZT, Ty 1 n! N=? (x +z)" w= aty=3 BBL 3 reer (EAS — artim = # Solln—9l n! Therefore we have — for arbitrary x, and x, — putting 2, + x, = 2,: ® xl @ xr > zr 2 ar Sw TS WE n=0 1" n=0 ""* n=0 *7* By our theorem, we have now established that the distributive law may at any rate be extended without change to infinite series, — and this, moreover, with an arbitrary arrangement of the products a,b, —, if both the two given series are absolutely convergent. It is conceivable that this restricting assumption is unnecessarily strict. On the other hand, the following example, given already by Cauchy?’ for the purpose, shows that some restriction is necessary, or the theorem no longer holds: Let . a,—=b,—=0 and a, — sai LT Ne por nl, so that X'q, and 2b, are convergent in accordance with Leibnitz’s rule 82,5. Then ¢,=¢,=0, and for n 2 2, 1 1 1 == or el n ( ) In vl Ee + =r] Replacing each root in the denominators by the largest, Vn —1, it follows that, for n > 2, n—1 le ve—1-yn—1 n n and therefore the product series X'¢, = 2 (ab, +a,0,_, + +++ + a,b) 27 Analyse algébrique, p. 149. 5 x Exercises on Chapter IV. 149 is certainly divergent in accordance with 82, 1. This is therefore a fortiori the case when we omit the brackets. Nevertheless, the question remains open, whether we may not be able, under less stringent conditions than that of absolute conver- gence of both the series 2'a, and Xb, to prove the convergence of the product series 2p, — at least for some special arrangement of the terms 4,b,, for instance as in the series X¢, above. To this question we shall return in § 45. Exercises on Chapter IV, 45. Examine the convergence or divergence of the series 3 (—1)"a,, for which a,, from some » onwards, has one of the following values: ry atn’ ant?’ Vr log n' log log n’ I Vn wo oi 46. What alterations have to be made in the answers to Ex. 34, when the behaviour of J (— 1)” a, is required? 47. Let JL tor 2381 «oy do2iiL, = k= ae En 1-1 tot 92kt1

] ple n=1 Yn a n—1 sponding to those of the series Fe in Ex. 50 and 51. When is the exactly corre- resulting series convergent and when is it not? When is the sum expressible in terms of the sum of the given series? 54. Consider, with the series 3 —, the same alterations in signs as in n : 1 : : : Ex. 52, for the series > When is a convergent series obtained? 85. For which values of « do the following two series converge: 56. The sum of the series lela aia, lies between 1 and 1, go ga 42 2 for every o> 0. 57. Given 0 1 = ol (-% show that 3 io i pegeie ol 1 2 1+ tr gba, : g : 1 1 1 1 l-prptptyp-grphtorang® (With the latter equality cf. Ex. 50c.) 58. Tn every (conditionally) convergent series the terms can be grouped together in such a manner that the new series converges absolutely. 58 a. The following complement to Kronecker’s theorem 82, 3 holds good: If a series Xa, is so constituted that for every positive monotone sequence (p,) tending to 4 ©, the quotients Pod +P +--+ Pata Pn tend to 0, then 2X a, is convergent. — In this sense, therefore, Kronecker's condition is necessary and sufficient for the convergence. § 18. The radius of convergence. 151 59. If from a given series Xa,, with the partial sums s,, we deduce, by association of terms, a new series 2X A; with the partial sums Sz, then the inequalities lim s, < lim S; < lim s,, invariably hold good, whether Xa, converges or not. 60. If Ya,, with the partial sums s,, diverges indefinitely, and s" is a value of accumulation (52) of the sequence (s,), then we can always deduce from Xa,, by association of terms, a series X 4; converging to s’ as sum. 61. If Xa,, with the partial sums s,, diverges indefinitely, and a, — 0, then every point of the stretch between the upper and lower limits of s, is a point of accumulation of this sequence. 62. If cvery sub-series of 3a, converges, then the series itself is absolutely convergent. 63. Cauchy’s product of the two definitely divergent series 22) ~ 6) lapels -(5 aa os) 2m) rd) La ; and is therefore absolutely convergent. How can this paradox be explained? and Chapter V. Power series. § 18. The radius of convergence. The terms of the series which we have examined so far were, for the most part, determinate numbers. In such cases the series may be more particularly characterised as having constant terms. This however was not everywhere the case. In the geometric series Xa", for instance, the terms only become determinate when the value of a is assigned. Our investigation of the behaviour of this series did not, consequently, terminate with a mere statement of convergence or divergence, — the result was: X'q¢" converges if|a| <1, but diverges if|a| > 1. The solution of the question of convergence or divergence thus depends, as do the terms of the series themselves, on the value of a quantity left undetermined — a wariable. Series which have their terms, — and accordingly their convergence or divergence, — depending on a variable quantity (such a quantity will usually be denoted by x 152 Chapter V. Power series. and we shall speak of series of variable terms) will be investigated later in more detail. For the moment we propose only to consider series of the above type whose generic term, instead of being a number q_, has the form n a, 2, 1. e.-we shall consider series of the form ota, xt a,x? + ela ha ces @® = Yaz" n=0 Such series are called power series (in x), and the numbers a, are their coefficients. For such power series, we are thus not concerned simply with the alternatives “convergent” or “divergent”, but with the more precise question: For what values of x is the series Convergent, and for what values divergent? Simple examples have already come before us: "1. The geometric series Xz" is convergent for |z| <1, divergent for || =1. For |z| <1, indeed, we have absolute convergence. TZ, 3 : 2. a is (absolutely) convergent for every real x; likewise the series 22k 2 2k+1 20 and Ze err nt 3. PES because For |z|>1, the series is divergent, because in that case (by 38, 1 and 40), n < |x|" is absolutely convergent for |z|<1. — +00. For x=1 it reduces to the divergent harmonic series, and for x =—1, to a series convergent by 82, Theorem 5. 0 n 4. Y > is (absolutely) convergent for |x |= 2, but divergent for rn n .9n T= 2] >2 : D. x nn" is convergent for x = 0; but for every value of x + 0 it is n=1 divergent, for if +0, |na|—>+ oo and a fortiori |n"z™|— +00, so that (by 82, Theorem 1) there can be no question of the series converging. For «= 0, obviously every power series Za a" is convergent, whatever be the values of the coefficients ¢,. The general case is evidently that in which the power series converges for some values of z, and diverges for others, while, in special instances, the two extreme cases may occur, in which the series converges for every x (Example 2), or for none + 0 (Example 5). ; 1. : : 1 The harmonic series 5 is also of this type: it converges for > 1, diverges for z < 1. 2 We here write, for convenience, x° =1 even when 2=0. § 18. The radius of convergence. 158 In the first of these special cases we say that the power series is everywhere convergent, in the second — leaving out of account the self-evident point of convergence x = 0 — we say that it is nowhere convergent. In general, the totality of points x for which the given series Xa, a" converges is called its region of convergence. In 2. this consists therefore of the whole axis of x, in 5. of the single point 0; in the other examples, it consists of a stretch bisected at the origin, — sometimes with, sometimes without one or both of its endpoints. In this we may see already the behaviour of the series in the most general case, for we have the OFundamental theorem. If Xa a" is any power series which 93. does mot merely converge everywhere or nowhere, then a definite positive number rv exists such that Za x" converges for every |x| <7 (indeed absolutely), but diverges for every |x| > vr. The number v is called the radius of convergence, and the stretch — v--- - » the interval of con- vergence, of the given power series®. — Fig. 2 schematizes the typical situation established by this theorem. CORY, ———2= 177574 -r 7 +r aw. Fig. 2. The proof is based on the following two theorems. OTheorem 1. If a given power series a,x" converges for © = x, (®, + 0), or cven if the sequence (a,x,") of its terms is only bounded there, then 2 a,x" is absolutely convergent for every x= x, nearer to the origin than x,, i.e. with |x, | <|#,|. Proof. X jo #|< K, say, then Zy Zo la, 2," | =|a,2," | CEY, ~ where 9 = the proper fraction 2 By 87,1 the result stated follows 0 immediately. OTheorem 2. If the given power series 2 a, a” diverges for x = x, then it diverges a fortiori for every x =x, further from the origin than zy, i.e. with |®, | > |x, | 3 In the two extreme cases we may also say that the radius of conver- gence of the series is # =0 or v = 4 00, respectively. 94. { 154 Chapter V. Power series. Proof. If the series were convergent for x,, then by theorem 1 it would have to converge for the point z,, nearer 0 than z,, — which contradicts the hypothesis. Proof of the fundamental theorem. By hypothesis, there exists at least one point of divergence, and one point of convergence = 0. We can therefore choose a positive number xz, nearer O than the point of convergence and a positive number y, further from 0 than the point of divergence. By theorems 1 and 2, the series 3g x” is convergent for z=1x,, divergent for x =1y,, and therefore we certainly have #, < y,. To the interval J, ==, ...y,, we apply the method of successive bisection: we denote by J, the left or the right half of J, according as Xa x" diverges or comverges at the middle point of J,. By the same rule, we designate a particular half of J, by J,, and so on. The intervals of this nest (J) all have the property that 3 a a" converges at their left end point (say z,) but diverges at their right end point (say y,). The number r (necessarily positive), which this nest determines, is the number required for the theorem. In fact, if x = 2’ is any real number for which |2’| <7 (equality excluded), then we have |2’| < z,, for a sufficiently large Z, i. e. such that the length of J, is less than » — |a’|. By theorem 1, 2’ is a point of convergence at the same time as x, is; and indeed at 2’ we have absolute convergence. If, on the contrary, 2” is a number for which |2”| > 7, then |a”| > y,,, provided m is large enough for the length of J to be less than |&”| —7. By theorem 2, 2” is then a point of divergence at the same time as y is. This proves all that was desired. This proof, which appeals to the mind by its extreme simplicity, is yet not entirely satisfying, in that it merely establishes the existence of the radius of convergence without supplying any information as to its magnitude. We will therefore prove the fundamental theorem by an alternative method, this time obtaining the magnitude of the radius itself. For this purpose, we proceed — quite independently of our previous theorem, — to prove the more precise OTheorem®: If the power series 2 a,x" is given and mu denotes the upper limit of the (positive) sequence of numbers lab imhe Vinh.» V7 Sale . p=1mV]a,]|, 4 Cauchy: Analyse algébrique p. 151. — This beautiful theorem remained for the time entirely unnoticed, till J. Hadamard (J. de math. pures et appl., (4) Vol. 8, p. 107. 1892) rediscovered it and made use of it in important appli- cations. § 18. The radius of convergence. 155 then a) if w=0, the power series is everywhere convergent; b) if w= 00, the power series is nowhere convergent; c) if 0 < u< +00, the power series converges absolutely for every |x| < Ly “ but diverges for every |x| > + Thus — with the suitable interpretation, > == lim V| a, | is the radius of convergence of the given power series’. Proof. If in case a) x, is an arbitrary real number == 0, then Fy > 0 and therefore by 89, 2] x, | Va | pis or |a 42h n 2], | n "0 on for every n >m. By 87, 1, this shows that Za x," converges ab- solutely, — which proves a). If conversely Xa x" converges for x = x, = 0, then the sequence na (a,«,") and, a fortiori, the sequence Cir % are bounded, If oo qo V] a, hd < K,, say, for every », then Vv] a, | = ne = K, for every u, 1 yo Le Wi a, ) is a bounded sequence. In case b), in which the sequence is assumed unbounded above, the series therefore cannot converge for any x == 0. 1 > Finally, in case c), if 2’ is any number for which |a'| < then choose a positive o for which |2/| < 0 < - and so > w. ‘By the definition of y, we must have, for every n > some 5,, Via] < : and consequently Via o"] << : <1. By 75,1, 2a, a'" is therefore (absolutely) convergent. ? ars : said 5 For convenience of exposition, we here exceptionally write — = 400, 0 1 ’ ; rum = 0. — Furthermore it should be noticed that ® i) is not for instance lim \/[ a, | the same as lim , — as the student should verify by means of obvious Vian] examples. (Cf. Ex. 24) 95. 156 Chapter V. Power series. On the other hand, if |” | > = so that 1 > < u, then we must have, for an infinite number of #’s (again and again; v. 59) nm 1 1 a,| >= 2’ By 82, Theorem 1, therefore the series certainly cannot con- verge. Thus the theorem is proved in all its parts. 0: |a,2"™|> 1. Remarks and Examples. 1. Since the three parts a), b), c) of the preceding theorem are mutually exclusive, it follows that the conditions are not merely sufficient, but also necessary for the corresponding behaviour of 3} a, 2”. nr 2. In particular, we have V | a, | => 0 for any power series everywhere convergent. For by the remark above, u=0, and since we are concerned with a sequence of positive numbers, these certainly have their lower limit # Zu. Since on the other hand » must be < 4, we must have x=p=0. nN By 63 the sequence (v Va: | ) is therefore convergent with limit 0 Thus for instance nil no spel, or nl—>00, xn because = oF converges everywhere. 3. Theorems 93 and 94 give us no information as to the behaviour of he series for z= + » and for & = — r; this differs from case to case: > a", io x : . : > =, > == all have the radius 1. The first converges neither at 1 nor at n n — 1, the second only at one of the two, the third at both. 4 Further examples of power series will occur continually in the course of the next paragraphs, so that we need not indicate any particular examples here, We saw that the convergence of a power series in the interior of the interval of convergence is, indeed, absolute convergence. We proceed to show further that the convergence is so pronounced as to be undisturbed by the introduction of decidedly large factors. We have in fact the 0 OTheorem. If >a, x" has the radius of convergence v, then the n=0 power series dma, x=, or what is the same thing, Sm+1a,,, x", n=0 n=0 has precisely the same radius. 6 Case c) may be dealt with somewhat more concisely: If Coif an : n lim V]a,| =u, then lim V|a,|+ |x| =1lim V]a, z"[=pu-|2| (for what reason?). By 76, 3 the series is therefore absolutely convergent for u-|x| <1, and certainly divergent for u-|2z|>1, q. e. d. § 18. The radius of convergence. 187 Proof. This theorem may be immediately inferred from Theo- rem 94. For if we write na, =a,’, then nn — n-——rvn __ Vie |=Vo Vu. Since (by 38, 7), Va—1. it follows at once from Theorem 62 that the sequences VT") and GT) have the same upper limits. For if we pick out the same sub-sequences from both, as corresponding terms only differ by the factor 1g which — - 1, these sub-sequences either both diverge or both converge to the same limit? Examples. 1. By repeated application of the theorem, we deduce that the series naar, In@m—Da,z"-2,....,Zu@w—1)..-m—k+ Da, z** or, what is exactly the same thing, the series : . Sm+Da yp a, Zm+l)n+2)a, 2", ..., k 2m+)yn42) -.. (+ Hanppar=n 3("T ) an ra” all have the same radius as Xa, 2", whatever positive integer be chosen for k. 2. The same of course is true of the series 2 rir, 0 Gn BLT Los rege Eten LR anti’ : ETRE IBS og INIT n : Thus far we have only considered power series of the form 2a, x". These considerations are scarcely altered, if we take the more general type 5 a, (= zy)" n=0 Putting # — x, = 2’, we see that these series converge absolutely for |@ | =|z—=,| <7, but diverge for | x — x, | > 7, if » again denotes the number deter- mined by Theorem 94. The region of convergence of this series — except in the extreme cases, in which it converges only for #=a,, or for every », — is therefore a streich bisected by the point z,, sometimes with, sometimes without one or both of its end- points. Except for this displacement of the interval of convergence, all our considerations remain valid. The point x, will for brevity be called the centre of the series. If x, = 0, we have the previous form of the series again. 7 Alternative proof. By 76, 5a or 91, 2, the series Znd*-1 is convergent for every |#| <1. If | a |< 7, and p is so chosen that zl <0 . (= = xo)" = 8 (=) n=0 and say that the power series defines, in its interval of convergence, a function of z. The foundations of the theory of real functions, that is to say the foundations of the differential and integral calculus, we assume, as remarked in the Introduction, to be already known to the reader in all that is essential. It is only to avoid any possible uncertainty as to the extent of the facts required from these domains, that we shall rapidly indicate, in the following section, all the definitions and theorems which we shall need, without going into more exact elucidations or proofs. § 19. Functions of a real variable. Definition 1 (Function). If to each value z of an interval of the x-axis, by any prescribed rule, a definite value y is made to cor- respond, then we say that y is a function of x defined in that interval and write, for short, : y=r(x) where “f” symbolises the prescribed rule in virtue of which each z has corresponding to it the relevant value of y°. The interval, which may be closed or open on one or both sides, bounded or unbounded, is called the interval of definition of f(x). Definition 2 (Boundedness). If there exists a constant K; such that for every x of the interval of definition we have fle) = K,, then the function f(x) is said to be bounded on the left (or below) in the interval, and K, is a bound below (or left hand bound) of f(x). If there exists a constant K, such that for every x of the interval of definition f(x) < K,, then f(x) is said to be bounded on the right (or above) and K, is a bound above (or right hand bound) of f(z). A function bounded on both sides is said simply to be bounded. There ‘then exists a constant K such that for every « of the interval of definition, we have - | f=) | < K. 8 Instead of an interval of the z-axis, we may also more generally take as basis a definite given set of points M of the z-axis. The function is then said to be defined in the set of points M. In the following §§ we shall however assume exclusively as basis intervals of the x-axis. § 19 Functions of a real variable. 159 Definition 83 (Upper and lower bound, oscillation). There is al: ways a least among all the bounds above of a bounded function, and always a greatest among all its bounds below?. The former we call the upper bound, the latter the lower bound, and their difference the - oscillation of the function f(x) in its interval of definition. Corre- sponding designations are defined for a sub-interval a’... 0d" of the interval of definition. Definition 4 (Limit of a function). If £ is a point of the interval of definition of a functicn f(x), or one of the endpoints of that interval, then the notation limflm)=q¢ >E or f@)—>¢c for z—£ means that a) for every sequence of numbers x, of the interval of definition which converges to &, but with all its terms different from &, the sequence of the corresponding values Yo =1%,) (=1,2,38,..9 of the function converges to c; or b) an arbitrary positive number ¢ being chosen, another positive number § = J (¢) can always be assigned, such that for all values of x in the interval of definition with - |e —&| <6 but z+§, we have LE) —e l= 22, The two forms of definition a) and b) mean precisely the same thing. Definition 5 (Right hand and left hand limits). If, in the case of definition 4, all points @, or x taken into account lie to the right of &, then we speak of a right hand limit (or limit on the right) and write : limzn)==c; z2->6+0 similarly we write imi y=='d, x>E-0 and speak of a left hand limit (or limit on the left), if points x, or x to the left of £ are alone taken into account. 9 Cf. 8, 2, and also 62. 10 The older notation lim f(z) for lim f(x) should be absolutely discarded v=§ z>& since the whole point is that x is to remain = &. 160 Chapter V. Power series. Definition 5a (Further types of limits). Besides the three types of limit already defined, the following may also occur: lim7(2)= or 6, =i, =—o4 f(x) — with one of the five supplementary indications (“motions of x”) for x—§& —E&4+0, —-&—-0, —+4o0, ——o0. With reference to 2 and 3 there will be no difficulty in formulating precisely the definitions — in the form a) or b) — which correspond to the definitions just discussed. Since, as remarked, we assume these matters to be familiar to the reader, in all essentials, we suppress all elucidations of detail and examples, and only emphasize that the value ¢ to which a function tends, for instance for x — &, need bear no relation whatever to the value of the function at &. Only for this we will give an evample: let f(x) be defined for every x by putting f(x) =0 if is an irrational number, but f(z) =~ if x is a rational number which in its lowest terms is of the form Lu> 0). Thus e. g. re =3,fO=f@=1,f({2=0, etc. Heve we have for every & z>& For if & is an arbitrary positive number and m is so large that Lar, then there are not more than a finite number of rational points whose (least posi- tive) denominator is < m. These we imagine marked in the interval £—1 ...&4+ 1. Asthere are only a finite number of them, we can find one nearest of all to &; (if & itself is one of these points we of course should nof take it into account here). Let § denote its (positive) distance from §. Then every x, for which is either irrational, or a rational number whose least positive denominator ¢ 3 : d 1 is > m. In the one case, f(x)=0; in the other, Elam Therefore we have, for every x in 0 < |2—&| <4, 1 f@-0]|§ i. e.,, as asserted, If therefore & is in particular a rational number, then this limit differs decidedly from the value f(&) itself. Calculations with limits are rendered possible by the following theorem: 1 In the first of these three cases we say that f(x) tends or converges to ¢; in the second and third cases: f(x) fends or diverges (definitely) to 4-00 or —00; and in all three, we speak of a definite behaviour or also of a limit in the wider sense. If ff (x) shows none of these three modes of behaviour, then we say that f(z) diverges indefinitely (for the motion of x under consideration) ~ . § 19. Functions of a real variable. 161 Theorem 1. If f, (x), fy(), ... f,(») are given functions (p some determinate positive integer), each of which, for one and the same motion of x of the types mentioned in Definition 5a, tends to a finite limit, say f, ( CT (2) —c,, then 2 the pits : f (x)= If, (6) + fo @) + rf, ett vrs b) the niin fl) =I[f@ f=) (EY) => Cty. oC c) in particular, therefore, the function af, (¥)— ac, , (a = arbitrary real number) and the function f, (¥) — f,(¥) —¢; — ¢,; , 1 . y d) the function =, provided ¢, 5-0. ke fi (@) Theorem 2. If limf (x) = ¢ (= + oo), then f(x) is bounded in a z>& neighbourhood of &, i. e. two positive numbers do and K exist such that |f(z)| < KE, when |z—§|<3, and corresponding statements hold in the case of a (finite) lim f (x) for xa—+£&-+0,%—0, +00, — 00. Definition 6 (Continuity at a point). If £is a point of the interval of definition of f(x), then f(x) is said to be continuous at & if lim f (x) z->& exists and coincides with the value f£(&) of the function at §: lim) = 1 (8) If we include the definition of lim in this new definition, we may also state: Definition 6a. f(z) is said to be continuous at a point &, if for every sequence of x's of the interval of definition, which tends to &, the corresponding values of the function A a Definition 6b. f(x) is said to be continuous at &, if, having chosen an arbitrary ¢ > 0, we can always assign é = d(¢) > 0, such that for every x of the interval of definition with 0<|z—¢&| 0) is continuous for every real z, log x is continuous for every x > 0, x (¢ = arbitrary real number) is continuous for every x > 0. Definition 8 (Continuity in an interval). If a function is conti- nuous at every individual point of an interval J, then we say that it is continuous in this interval. The endpoints of J may or may not, according to the circumstances, be reckoned as in the interval - Functions which are continuous in a closed interval give rise to a series of important theorems, of which we may mention the following: Theorem 4. If f(x) is continuous in the closed interval a 0, but f(b) < 0, then there exists, between a and b, at least one point & for which f(£) =0. Theorem 4a. If f(x) is continuous in the closed interval a 0, we can always assign some number 0 > 0 so that, if 2’ and 2” are any two points of the interval in question whose distance |2” — a’| is < J, the difference of the corre- sponding values of the function, |f(2”) — f(2’)|, is f(x,), in the other. We also speak of strictly increasing and strictly de- creasing functions, when the equality signs, in the inequalities between the values of the function just written down, are excluded. Theorem 6. The point &, certainly existing under the hypotheses of Theorems 4 and 4a, is necessarily unique of its kind if the func- tion f(x) under consideration is strictly monotone in the interval a...b. Thus in that case, to each #3 between f(a) and f(b) corresponds one and only one & for which f(&)=#. We say in this case: The inverse function of y= f(x) is everywhere existent and one-valued (or y = f(x) 1s reversible) tn the interval. Definition 10 (Differentiability). A function f(x) defined at a point & and in a certain neighbourhood of & is said to be differentiable at & if the limit : 1-1 ® ai CE exists. Its value is called the (unique derivative or) differential coefficient of f(x) at £ and is denoted by f’(£). If the limit in question only exists on the left or on the right (that is, only for x—& 4-0 or x— & — 0 respectively), then we speak of right hand or oy hand differentiability, differential coefficient, etc. If a function is differentiable at each individual point of an inter- val J, then we say for brevity that the function is differentiable in this interval. The rules for differentiation of a sum or product of a particular (fixed) number of functions, of a difference or quotient of two functions, of functions of a function, as also the rules for differentiation of the elementary functions and of their combinations, we regard as known to the reader. All means necessary to their construction have been developed in the above, if we anticipate a knowledge of the limit defined in 112 and there determined in a perfectly direct manner, If, for instance, it is inquired whether a” (a> 0 and = 1) is differentiable, and, if so, what is its differential coefficient, at the point &, then, following Defs. 10 and 4, we have to choose a null se- quence (z,) with terms all 4 0 and to examine the sequence of numbers r dent a®n — 1 Ln He TARE If we write y, for the numerator in the last fraction, then by 835, 3 we know that (y,) is also a null sequence, and indeed one for which none of the terms is equal to 0. X, may then be written in the form : -log a X, wai dn : : log (1+ v5) But since, as remarked, y, is a null sequence, we have by 112 log (1+v,) — 1. Yn Since the same then holds for the reciprocal values, by 41, 11a, we deduce X, — at.log a. The function a” is thus differentiable for every x and has the differential coefficient a®.log a. n= - 164 Chapter V. Power series. In the same way, as regards differentiability and differential coefficient of log x for £ > 0, we deduce, by consideration of Xa lo +72) 3 _logtay-togg “EVYE/ fie > 4 § z, %a 3 that the differential coefficient exists here and a & Of the properties of differentiable functions we shall for the pre-— sent require scarcely more than is contained in the following simple theorems: Theorem 7. If a function f(z) is differentiable in an interval J and its differential coefficient is there constantly equal to 0, then f(x) is constant in J, that is to say is ={f(x,), where z, is any point of J. If two functions f, (x) and f, (x) are differentiable in J and their differential coefficients constantly coincide there, then the difference of the two functions is constant in J, therefore we have fo@)=f,(x) +c=1,@) + [1 (%) — £,(x,)] where x, is any point of J. Theorem 8. (First mean value theorem of the differential calculus.) If f(x) is continuous in the closed interval 4 0 (<0) then f(x) “increases” (“decreases”) at §, i. e. the difference the same : f(x) — f(&) has 0 Pppotiie) | sign as (to) (x — &), provided |x — &| be less than a suitable number 4. Theorem 10. If f(x) is differentiable at &, then unless f{H=0 the functional value f(&) cannot be > every other functional value f(x) in a neighbourhood of & of the form |x — &]| <8, 1. e. £ cannot be a maximum point. Similarly the condition f’(&§)= 0 is necessary for £ to be a minimum point, i. e. such that f(£) is not greater than any other functional value f(x), as. long as x remains in a suitable neighbourhood of §. Definition 11 (Differential coefficients of higher orders). If f(z) is differentiable in J, then (in accordance with Def. 1) f’(z) is again a function defined in J* If this function is again differentiable in J, * and called the derived function of f(x). - § 19. Functions of a real variable. . 165 then its differential coefficient is called the second differential coefficient of f(x) and is denoted by f” (x). Correspondingly, we obtain the third and, generally, the kt differential coefficient of f(x), which is denoted by f® (x). For the existence of the kt differential coefficient at & it is thus (v. Def. 10) necessary that the (k — 1)% differential coefficient should exist both at& and at all points of a certain neighbourhood of &. — The jth differential coefficient of f* (x) is f*+0 (x), k>0, 1 >0. (As Ott differential coefficient of f(x) we then take the function itself) Of the integral calculus we shall, in the sequel, require only the simplest concepts and theorems, except in the two paragraphs on Fourier series, where rather deeper material has to be brought in. Definition 12 (Indefinite integral). If a function f(x) is given in an interval g...b and if a differentiable function F (x) can be found such that, for all points of the interval in question, F’(x) = f(x), then we say that F(x) is an indefinite integral of f(x) in that interval. Be- sides F(x), the functions F(x)-}-¢ are then also indefinite integrals of f(x), if ¢ denotes any real number. Besides these, however, there are no others. We write F@)=[f@)de. In the simplest cases, indefinite integrals are obtained by inverting the elementary formulae of the differential calculus. E. g. from (sin « @) = « cos ax ; sin o¢ x it follows that [eosawaz= —, and so on. These elementary rules we a assume known. Special integrals of this kind, excepting the very simplest, are little used in the sequel; we mention 22-1 dx 1 1 1 [TEm=grstta-gura-—rtans twas 202L, a : hs = i | dx _y?2 ow +z V2 41 ve Corie 2 8 2p yl] l—2 14-24 [[eota—1] dz= bg 22, x x Though in indefinite integrals, we find no more than a new mode of writing for formulae of the differential calculus, the definite integral introduces an essentially new concept. Definition 13 (Definite integral). A function defined in a closed interval a... b and there bounded is said to be integrable over this interval if it fulfils the following condition: Divide the interval ¢...b in any manner into # equal or un equal parts (n > 1, a positive integer), and denote by =z, @,,...,, _, the points of division between a =u, and b =x,. Next in each of these » parts (in which both endpoints may be reckoned) choose any . \ 166 Chapter V. Power series. point, and denote the chosen points in corresponding order by &,, £,2+.4:%,." Then form the sum = (=, a) i 2) oi? Let such sums S, be evaluated for each n=1, 2, 3, ... independently (that is to say, chosing #, and &, each time afresh) but seeing, at the same time, that / , the length of the longest of the n parts into which the interval is divided when forming S, , tends to 03. ; If the sequence of numbers S,, S,, ..,, in whatever way they may have been formed, invariably proves to be convergent and always gives the sams limit S*, then f(x) will be called integrable in Riemann’s sense and the limit S will be called the definite integral of f(x) over a ...b, and written S$ = [r@ dx x is called the variable of integration and may of course be replaced by any other letter. — Instead of f(£) we may also take, to form S , the lower or the upper bound of all the functional values in the inter- val ov, 5... 0085. Theorem 11 (Riemann’s test of integrability). The necessary and sufficient condition for a function f(x), defined in the closed interval . b and there bounded, to be integrable over a... b, is as follows: Given 2 > 0, a choice of » and of the points #,, #,, «.., # must be possible, for which n=—21 n >ho, 0, a>b, and we consider a plane portion S bounded on the one side by the axis of abscissae, on the other by the verticals through a and b and by the curve y = f(z), then S, is an approximate value of the area of S. This however only provides a satisfactory representation if y = f(z) is a curve in the intuitive sense. 13 We may then also say that the subdivisions, with increasing #, become indefinitely closer. 14 It is easily shewn that if the sequence (S,) is invariably convergent it also ipso facto always gives the same limit. 15 In these cases S, gives the area of a polygon inscribed or gireumecringd to the plane portion S. § 19. Functions of a real variable. 167 Riemann’s test of integrability may also be given the following form: _— Theorem 14. The function f(x) is integrable over a... b if, and only if, it is bounded there and if its points of discontinuity — whether Dy finite or infinite number — given an arbitrary number ¢ > 0, can always be enclosed in a finite or infinite number of intervals of total length less than gS. Theorem 15. The function f(x) is certainly not integrable over .b if it is discontinuous at every point of that interval. Theorem 16. If f(x) is integrable over a...b, then f(z) is also integrable over every sub-interval a’... 5" of a...b. Theorem 17. If the function f(x) is integrable over a...b, then every other function f, (x) is integrable over a... b, and has the same integral, which results from f(x) by an arbitrary change in a finite number of its values. : Theorem 18. If f(z) and f, (x) are two functions integrable over a...b, then they have the same integral provided that they coincide at least at all points of a set everywhere dense in a4...5 (e. g. all rational points). For calculations with integrals we have the following simple theo- rems, where f(z) denotes a function integrable over the interval a...b. a b Theorem 19. We have [1 dz = — [ f(®)de and if a, dpr ds b a are three arbitrary points of the interval a... b, Qo ag ay [f@de+ [f@) da + [ f@)dz=0. a; Oo as Theorem 20. If f(x) and g(x) are two functions integrable over a...0,{a 0 or the upper bound < 0, then re is also integrable over a... b. Theorem 23. If f(x) is integrable over a... b, then the function x F(@)= [f(a a is continuous in the interval a... b and is also differentiable at every point of the interval, where f(x) itself is continuous. If x, is such a point, then F’(z,) = f (x,) there. Theorem 24. If f(x) is integrable over a...b, and F(x) is an indefinite integral of f(x) in that interval, then J f@) dz = F(b) — F(a) (Principal formula for the evaluation of definite integrals). Theorem 25 (Change of the variable of integration). If f(x) is integrable over ¢...b and x = ¢ (¢) is a function differentiable in «... 3, with ¢(¢) =a and ¢(f) =1b, if further, when ¢ varies from « to f, @ (t) varies monotonely (in the stricter sense) from a to b, and if ¢'(2), the differential coefficient of ¢ (f), is integrable over «... 17, then b 8 ne dw =[f(p0)-# (¢) as. Theorem 26 (Iniegration by parts). If f(x) is integrable over a...b and F(x) is the indefinite integral of f(z), if further g(x) is 17) The derivative of a differentiable function need not be integrable. Examples of this fact are, however, not very easily constructed (cf. e. g. . H. Lebesgue, Legons sur l'intégration, Paris 1904, pp. 93—94). § 19. Functions of a real variable. 169 a function, differentiable in a... bd, whose differential coefficient is inte- grable over 4... 0, then b Jf @)e(@)de = [F (2) g (x) — [F(9)-¢ @ az. 18 The following penetrates considerably further than all the above simple theorems: Theorem 27 (Second mean value theorem of the integral calculus). If f(x) and (x) are integrable over a... b and ¢ (2) is monotone in that interval, then there exists a number §, with a < & a z->b .We mention only the following of the applications of the concept of integral above considered: Theorem 28 (Area). If f(x) is integrable over a...b, (a a and is integrable over aL tL x, for every z, so that the function F@= [f(a is also defined for every x > a, then we say that the improper integral Sra converges and has the value ¢, if lim F() exists and =. zT>+ oo Theorem 80. If f (t) is constantly > 0 or constantly <0 for every ¢ > a, then f f (t)dt converges if and only if the function F (x) : ; of Def. 14 is bounded for « > a. If f(f) is capable of both signs for ¢{ >a, then the same integral converges if, and only if, given an arbitrary ¢ > 0, 2, > a can be so determined that [roa =,. And quite analogously: Definition 15. I f (x) is defined in the interval a < ¢ < b, open on the left, and is integrable, for every x of a < x a+0 Theorem 31. If in the case of Def. 15, we further have f(£) > 0 everywhere or < 0 everywhere, then the improper integral in question exists if and only if, F (x) remains bounded in a 0, we can choose 6 > 0 so that welt z [frat] 0), but may be - co, i. e. the series may be everywhere convergent. — We then have, first, the OTheorem. The function f(x) defined, in its interval of conver- 96. [ee] gence, by the power series 3 a, (x — x)", is continuous at x = xy; n=0 that is to say, we have lim. /(n) = in > n,n — oY =a, =f (n,). T—>% z>72, n=0 Proof. Kf. 0. < p< 7, then by 88, 5, 5 |a,|on~! converges with Slalom If we write "K(> 0) for the sum of the ford, then we have, for every |x — z,| 0 is arbitrarily given and if ¢ > 0 is less than both & ¢ and —, then we have, for every |z — z,| < J, |f (@) — an] <&; which by § 19, Def. 6b, proves all that was required. From this theorem, we immediately deduce the extremely far- reaching and very frequently applied: Oldentity Theorem for power series. If the two power series x a,%" and 3 b* n=0 n=0 both have a radius of convergence = po > 0 (this number @ may, for the rest, be as small as we please), and have the same sum for every |@| < 0, then the two series are entirely identical, that is to say, for every n=20,1,2,..., we then have Proof. From a=, (a) t,t axa, Bt =b bt Bat Lee it follows, by the preceding theorem, letting x — 0 on both sides of the equation, that a, = b,- Leaving out these terms and dividing by @, we infer that for 0 < || <0 (b) a, +a,zta,2®t..-=b }bx-tba?t}--., an equation from which we deduce, in exactly the same way??, that a, =b Ota xt =0, Bre Proceeding in this manner, we infer successively (more precisely: by complete induction) that for every mn the statement is fulfilled. and Examples and illustrations. 1. This identity theorem will often appear both in the theory and in the applications. We may also interpret it thus: if a function can be re- presented by a power series in the neighbourhood of the origin, then this is only possible in one way. In this form, the theorem may also be called the theorem of uniqueness. It of course holds, in the corresponding statement, for the general power series Xa, (x — xy)" 2. Since the assertion in the theorem culminates in the fact that the corresponding coefficients on both sides of the equation (a) are equal, we may also speak, when applying the theorem, of the method of equating coefficients. 21 Or even for every x=u2x, of a null sequence (zy) only. — In the proof we have then to carry out the limiting processes in accordance with § 19, Def. 4a. 2 For x=0, equation (b) is not in the first instance secured, since it was established by means of division by z. But for the limiting process z— 0 this is quite immaterial (cf. § 19, Def. 4). '§ 20. Principal properties of functions represented by power series. 173 3. A simple example of this form of application is the following: We certainly have, for every =z, (+a) (+a) =1+2)2* = or Zhe 20)--2 00 If we multiply out on the left, by 91, Rem. 1, and equate the coefficients on both sides, then we obtain, for instance, by equating the coefficients of x: EG) reed) (=x (Fe (3), a relation between the binomial coefficients which would not have been so easy to prove by other methods. 3 4. If f(x) is defined for |x| OTheorem 3. A function represented by a power series re) = Sa, — a n= is differentiable at every interior point x, of the interval of convergence 23 We thus have, quite incidentally, a fresh proof of the convergence, already established in 995, of the different series obtained for the coefficients by. § 20. Principal properties of functions represented by power series. 175 (v. § 19, Def. 10) and its differential coefficient at that point, f’(,), may be obtained by means of term-by-teym differentiation, i. e. we have fn) = Sma, (y= = 80+ aya (oy = 50)" n= n= Prooi. Since f{z)=— 5 (x — x,)", we have for every x suf- n=0 ficiently near x, tre) =, by (2 — 2) + + whence for z—2,, by 96, taking into account the meaning of b,, we at once deduce the required result: f/(2,)=b, = J na (v, — zr Theorem 4. A function represented by a power series, fia) == 2 a,x — u,Y, n= has, at every interior point x, of its interval of convergence, differential coefficients of every order and we have Fr Y= 210, == Se +1)(n+2)...(n+k)a, (2 —2)" Proof. For every x of the interval of convergence we have, as: we have just shown, f@)=23 n+) — a)" = f'(z) is thus again a function represented by a power series, — and in fact by one which, in accordance with 93, has the same interval of convergence as the original series. Hence the same result may be again applied to f' (x), giving {’@) = 5 n(n Dag (@® — 2) = =3@ +1) (n+ 2) apis (® — ,)" By a ein gf this simple process, we obtain for every k, f® (x) = 3 (n+ Din +2)... (0 +k) (a, 4, @ — 2)", — valid for Sher x ®t the original interval of convergence. Putting in particular # = ,, we therefore at once deduce the required statement. If we Tae for the coefficients b, in the expansion of theorem 1, the values =F (ey) now obtained, then we finally infer from all the above the so-called ©Taylor series?*. If for J x,| <7, we have fx) = Sa, (@ — a)", and if ® is an interior rt ¥ the interval of convergence, then we # Brook Taylor: Methodus incrementorum directa et inversa, London 1715. -— Cf. A. Pringsheim, Geschichte des Taylorschen Lehrsatzes, Bibl. math. (3) Vol. 1, p. 433. 1900. 176 Chapter V. Power series. have, for every |x — x, | [ONE+1 1\* 2s - ree SF er)) k=0 and the radius of this series is not =7— |, —2,| = L but is 3. § 20. Principal properties of functions represented by power series. [77 1 : : : ; of sum {=> We can deduce from our considerations neither its con- — tinuity at the point x = — 1, nor its discontinuity at x = -}1, by immediate inspection of the series. Even if the power series con- verged at one of the endpoints of the intervals (as here 3 for r= — i, we should not be able to conclude this fact directly. That however, in this last particular case, the presumption is, at least to some extent, justified, we learn from the following: Abel's limit theorem?®. Let the power series f(@)= > a 2" 100. n=0 have radius of convergence v and still converge for x = |r. @D Then Hm f(x) exists apd «=X a, r2. x>r—0 n=»0 Or in other words: If > a a" still converges for x= +r, then n=0 the function f(x) defined by the series in —r 1-0 n=J Now by 91 (v. also later, 10%), we have for |z| <1, 1 3 oa O 1—= 2 @, x" gre, a" = > Sn if > if by s, we denote the partial sum of X'a,. Consequently f(x) = (1—2x) 2's, 2" and since 1=(1—x) 2a", we therefore deduce, for j=] <1, (a) s— f(x) =(1— 2) 3 (s —s5,)"=01-— 2) 3, 2. 26 Journal f. d. reine u. angew. Math. Vol. 1, p. 311. 1:26. cf. 283 and § 62. — The theorem had already been stated and used by Gauss (Disquis. generales circa seriem ..., 1812; Werke III, p. 143) and in fact precisely in the form proved further on, that 7, — 0 involves (1 —2) 7,2" > 0 if 2 — 1 from the left (v. eq. (a)). The proof given by Gauss loc. cit. is however in- correct, as he interchanged the two limiting processes which come under con- sideration for this theorem, without at all testing whether he was justified in so doing # This remark holds in general for all discussions of (not everywhere convergent) power series of positive radius #. 178 Chapter V. Power series. Here we have written s — s, = 7,, the “remainder” of the series; these remainders, by 82, Theorem 2, form a null sequence. ; If now a positive ¢ < 11s arbitrarily given, then we first choose Ue i A Sb sa Ris Ll m so large that, for every n> m, we have |7, | < 3 We then have. for 0 5 21, m @® s—f@)| £10 — 2) Ira" | +50 — 2-3, n= n=m+1 hence if we put ? |ldltsh etl, |= —7+0 n=0 Remarks and examples for these and the following theorems of the present paragraph will be given in detail in the next chapter. : The continuity theorem 98, 2 and Abel's theorem 100 together 3 assert that 101. im( St) = Yu, i ax—>E i if the series om the right converges and x tends to & from the side on which lies the origin. If the series Ja, &" diverges, we cannot assert anything, without further assumptions, as to the behaviour of Xa, a" when z—&. We have however in this connection the following somewhat more definite: Theorem. If Xa, is a divergent series of positive terms, and 2a, x" has radius 1, then f(x) = Sa," + co when x tends towards —1 ow the origin. Proof. A series of positive terms can only diverge to A-oo! If therefore G > 0 is arbitrarily given, we can choose m so large that a,~+ a, + ---+a,>G-41, and then by §19, Theorem 3, choose 0 <1 so small that for every 1 >2>1— 6, we continue to have ay+a, z+ -+a,x™>G. § 21. The algebra of power series. 179 But then we have, a fortiori, f(@) = 2a," = G, n=0 which is all that required proof. § 21. The algebra of power series. Before we make use of the farreaching theorems of the preceding section (§ 20), which lead to the very centre of the wide field of appli- cation of the theory of infinite series, we will enter into a few questions whose solution should facilitate our operations on power series. That power series, as long as they converge, may be added and subtracted term by term, already follows from 83, 3 and 4. That we may immediately multiply out term by term, in the product of two power series, provided we remain in the interior of the intervals of convergence, follows at once from 91, since power series always converge absolutely in the interior of their intervals of convergence. We therefore have, with Za ut hak mela i Yo PD oo 0 also Ya of 3b o"=2ab +a,b. t+ ab) n=0 n=0 n=0 provided xz is interior to the intervals of convergence of both series ?. The formulae 91, Rem. 2 and 3 were themselves a first appli- cation of this theorem. IH the second series is, in particular, the geometric series, then we find 2 n 7 z vd Your = Ysa, n=0 n=0 n=0 7i8 Il 1. e, 1 n a n > a, xz = £2 S$, n=0 n=0 Sajon=(1—ux) Ys, x", n==0 n=) 1—= or where s =a, a,+ +--+ +a,, and |#| <1 and also less than the radius of Sa 2". We infer in as simple a manner that every series may be mul- tiplied — and in fact, arbitrarily often — by itself. Thus @® (2 0a") =X (aya, + a,a,_, ++ + a,4)a% n=0 and generally, for every positive integral exponent g, 2 ” S (Kk) Ya," =n." x" n=_ n=y90 28 Here we see the particular importance of Cauchy's product (v.91, 1) 102. 103. 104. 180 Chapter V. Power series where the coefficients a’ are constructed from the coefficients a,ina perfectly determinate manner — even though not an extremely obvious one?? for larger k's. And these series are all absolutely convergent, so long as Xg a" itself is. This result makes it seem probable that we “may’ ’ also divide by power series, — that for instance we may also write 1 2 se = x p= cee atin Tart Ta and that the coefficients ¢, may again be constructed in a perfectly determinate manner from the coefficients @,. For we may first, writing — or et a); forn=1,2,3,..., replace the left hand ratio by 0 3 1 1-2 va ot) and then by 1 2 Lihat aa +o) hoe aa] which must actually result in a power series of the form X¢ 2", if the powers are expanded by 103 and like powers of x then grouped together. Our justification for writing the above may at once be tested from a somewhat more general point of view: We suppose given a power series 2g 2" (in the above, the series Xa x), whose sum we denote by f(x) or more shortly by y. n=0 We further suppose given a power series in yy, for instance g(y)=20b,y" (in the above, the geometric series X'y") and in this we substitute for y the former power series: bob Uy (2 #4 Ere YF Olay FR, BAF nen, Under what conditions do we, by expanding all the powers, in accordance with 108, and grouping like powers of x together, ob- tain a new power series ¢, +c, * 4 ¢,* 4---- which converges and has for sum the value of the function of a function g(f(x))? We assert the i 8 = 2 OTheorem. This certainly holds for every x for which Il © n converges and has a sum less than the radius of Z'b,y". 20 Recurrence formulae for the evoluation of a® are to be found in J- W. L. Glaisher, Note on Sylvester's paper: Development of an idea of Eisen- stein (Quarterly Journal, Vol. 14, p. 79—84. 1175), where further references to the bibliography may also be obtained. See also B. Hansted, Tidskrift for Mathematik, (4) Vol. 5, pp. 12—16, 1881. : % Hh Eh SC CB Ca de § 21. The algebra of power series. 181 Proof. We have obviously here a case of the main rearrangement theorem 90, and we have only to verify that the hypotheses of that theorem are fulfilled. If we first write Ve (ag, Lat Vf=a® 1 aPal 0,001 i forming the powers by 103, and also suppose this notation adopted for k=0 and k = 13°, then we have, in bp =0,(2> Lan. > Lg I...) be ir Era heb ks > (a) : i ‘ by =0b (a or Pi wk brs ) the series z¥ occurring in Se diecrem 90. If we now take, instead of y=Z4q 0", the series y= 2a o*|, and, wiiling [z|=2, form, quite similarly, de ratiabe i) ; njr={rend in i TL ) . . . Ilo" =I el tls #40) : (A) then all the Shu in this array (A) are > 0 od since en |b, | 7 was assumed to converge, the main rearrangement theorem is srulicatle to (A’), But obviously every number of the array A is in absolute value < the corresponding number in (A'); hence our theorem is a fortiori applicable to (A) (cf. 90, Rem. 3). In particular, therefore, the coefficients standing vertically one below the other in (A) always form (absolutely) convergent series ww Sha P=0¢, {for every definite. #=0,1,2,..) k=0 and the power series formed with these numbers as coefficients, i. e. @D 20 ak n=0 is again, for the considered values of x, (absolutely) convergent and has the same sum as > b, y". We therefore have, as asserted, sf @) = 2c, = n= with the indicated meaning of c,. Remarks and Examples. 105. 1. If the “outer” series g(¥)= 2brpa* converges everywhere, then our theorem evidently holds for every x for which Xa, 2" converges absolutely. 30 We have therefore to write al® =1, a® = 2 =ere=0, and a =a,, the latter for n=0,1,2,.... 7% 182 i ~ Chapter V. Power series. - If both series converge eoeryaher, then the theorem holds without restriction for every z. s 2. If a; =0 and both series have a positive radius, then the theorem certainly holds for every “sufficiently” small x, that is to say, there is then certainly a positive number p, such that the theorem holds for every |z | 0, we now also, by 96, have » — 0, 5 is certainly less than the radius of 3b, 9% for all x whose absolute value is less than a suitable number p. n 3. In the series zr , we “may” for instance substitute y =X a" for Zk : [2] <1, or y= SIE for every mn, and then rearrange in powers of x. 4. To write, as we did above: : 1 a, +a, x+a, x24. is, we now see, certainly allowed if a, & 0 and further x is in absolute value _so small that =CotCc x4 Coa? oon sia ll al... ir 5g] + | Zar] + 2h which by Rem. 2 is certainly the case for every |x|

a,cy=1 a,c, +a, c,=0 ayCot+a,c,+ay,co=0 Qo Cy + ay Cot ay0, +a5c,=0 Sie Site mite te of ey vu — from which, since a, 3 0, the coefficients cy, ¢,, cp, ... may be uniquely determined in succession?2, 5. As a particularly important example for many subsequent investigations we may set the following question?3: Expand 115 22 or a, 23 ) 1 SBE mY ((+e+ 2454.) 81 How small x has to be, is usually immaterial. But what is essential, is that some positive radius p exists, such that the relation holds for every |z| +} --- in the form (a, x) +o (a 2° ai (02 + ro. in the numerator, a number with 107 digits. The numbers B,, B,,..., to By, had previously been calculated by Oam, ibid, Vol. 20, p. 111, 1840. — The numbers B,, first occur in james Bernoulli, Ars conjectandi, 1713, p. 96. — A comprehensive account is given by L. Saalschiitz, ,,Vorlesungen iiber die Bernoullischen Zahlen®, Berlin (J. Springer) 1893. New investigations, which chiefly concern the arithmetical part of the theory, are given by G. Frobenius, Sitzgsber. d. Berl. Ak., 1910, p. 809—847. 85) Or: we write for brevity x —ax,=2' and y —y,=3" and then, for simplicity’s sake, omit the accents. bi id GiB Ca ad § 21. The algebra of power series. 185 If we write for brevity a, =a’ and, for n > 2, and subsequently, for simplicity’s sake, omit the accents, then we obtain precisely the above form of expansion. It suffices therefore to consider this. But we can then show that a power series, convergent in a certain interval, of the form (b) we==y | B90, 9 + ves exists which represents the inverse function of the former, so that © tbr + Yt tb, Pte Fla (yl 592 eB Le is identically = v, if this series is arranged in powers of y, in accor- dance with 104, — 1i e. all the coefficients must be = 0 except that of 4, which is ==1. Since we have written, for brevity, # instead of a, x, we see that the series on the right hand side of (b) has still to be divided by a, to represent the inverse of the series a, x | 4,2*-}--.-, where 4, has no specialised value. In this general case we shall therefore have . b= as coefficient of y. If we assume, provisionally, that the statement (b) is correct, then the coefficients b, are quite uniquely determined by the condition that the coefficients of 92% 4% ... in (c) after the rearrangement, have all to be = 0. In fact, this stipulation gives the equations b, + a,=0 (d) b, | 20, ay + ag = 0 b, + (b> + 2b) a, + 3b,a, +a, =0 from which, as is immediately evident, the coefficients b, may be determined in succession, without any ambiguity. Thus we obtain, the values by, = — a, by = —2b,a, — a, = 2a,’ — a, ©) b= — (0, + 2b;)a, — 3b,0;, — a, Dams but the calculation soon becomes too complicated to convey any clear idea of the whole. Nevertheless, the equations we have written down show that if there exists at all an inverse function of y= f(x), capable of expansion in form of a power series, then there exists only one. ’ Now the calculation just indicated shows that whatever may have been the original given series (a), we can invariably obtain perfectly 186 Chapter V. Power series. determinate values b,, so that we can invariably construct a power series y --b, y® ----- which at least formally satisfies the conditions of the problem, the series (c) becoming identically =y. It only remains to be seen whether the power series has a positive radius of convergence. If that can be proved, then the reversion is Somileny carried out. The required verification may, as Cauchy first showed, actually - be attained, in the general case, as follows: Choose any positive numbers ¢, for which we have : | a,| = «, and X ¢, x» has a positive radius of convergence. Proceeding in the above manner, for the series: y= — a, 2% — a; 2 - - whose inverse is, then, say, RE Yt fo? Lf 0 een we obtain, for the coefficients fg, the equations : B, = — Gy / fy 2, i +o fl 3 -2h,)e B55 2 in which all the terms are now positive. Thus we every », B, = | b |. If, therefore, it is possible so to choose the ¢, that the series Xf y” has a positive radius of convergence, it would follow that Xb 3” also had a positive radius and our proof would be complete. We choose the «’s as follows: There is certainly a positive number p, for which the original series x + a,2®-} --- converges absolutely. A positive number K must, however, then exist (by 82, Theorem 1 and 10, 11) such that we have, for every v=2,3,..., lol" 0, since the second is and the two Sire product = p% But I nl lr [im 2f] 2(K+o) e 91 y/ 4° In the following chapter we shall see that, for |z| <1, the power Qt can actually be expanded in a power series — beginning with 1— 5+ . Assuming this result, it follows immediately that x also may be expanded in a power series, convergent at least for lyl <3: y v= satry 1+ Poland wo) (t=) jimmy ol yf or By our first remarks the phon? is hereby entirely completed. The actual construction of the series y+ by" + Zt a, xt. here also involves in general considerable difficulties and necessitates the use of special artifices in each particular case’. Examples of this will occur in §§ 26, 27. ; We only note further, a fact which will be of use later on, — that if (b) is the inverse of (a), then the inverse of the series (@) y= — a,x a, ® — +f —... where the signs are alternated, is obtained from (b) by similarly alternating the signs, i. e. (b') 2 =y—>5yy Fle ies - from the series 36 The general values of the coefficients of expansion b, are worked out as far as b,, by C. E. van Orstrand, Reversion of power series, Philos. Magazine (6), Vol. 19, p. 366, 1910. 138 . Chapter V. Power series. This is at once evident, if we first actually expand the powers of (y +85" +...) in (c), obtaining, say, : (© ht Yl a ERNE : Ta Eh, oe) Lee, Under the new assumption, the same process, since the product of two series with alternating coefficients is again a series with alter- nating coefficients, gives (©) 0=5,7 +=) =a. {y = 5,5 Ls.) 1000 —5,y | Vruee, _ And from this we immediately infer that on equating to zero the coefficients of 2, y3,..., we must obtain the identical equations (d), thus deducing for b, precisely the same values as before. Exercises on Chapter V. 64. Determine the radius: of convergence of the power series Xa, 2", when a, has, from some point onwards, the values given in Ex. 34 or 45. 65. Determine the radii of the power series 1.2 on . ie: 2 Soot. MCC) Zag) Ih I a1; ope an? CDI n 66. Denoting by x» and pu the lower and upper limits of , the Ant1 radius 7 of the power series Xa, x" invariably satisfies the relation x (an + a; nx, 4, a a, En 67a. What is the radius of Ba if 01-0 0 (n= Fay Feet). § 22. The rational functions. 189 ¢ 71. The converse of Abel's theorem 100, not in general true, holds, however, if the coefficients a, are = 0; if therefore, in that case, lim Ya, zx” z->r—0 exists, then Xa, #” converges and its sum is equal to that limit. v3. Let REE © Sa =fE and Db, 2*=g(), n=1 n=1 both series converging for |z| are in that case all = 0. Formulae such as those we have just deduced have — as we may observe immediately, and once for all — a two-fold meaning; if we read them from left to right, they give the expansion or representation of a function by a power series; if we read them from right to left, they give us a closed ex- pression for the sum of an infinite series. According to circumstances, the one interpretation or the other may occupy the foremost place in our attention. By means of these simple formulae we may often succeed in expanding, in a power series, an arbitrary given rational function a aya, x4. +a,z™ i= bob, x +--+ bak’ namely whenever f(z) may be split up into partial fractions, i. e. ex- pressed as a sum of fractions of the form 4 (@— a)? Every separate fraction of this kind, and therefore the given func- tion also, can be expanded in a power series by 108. And in fact this expansion can be carried out for the neighbourhood of every point , distinct from a. We only have to write trill a—x, § 23. The exponential function, 191 and then expand the last fraction by 108. By this means we see, at the same time, that the expansion will converge for |x — x, | <|a— | and only for these values of x. This method, however, only assumes fundamental importance when we come to use complex numbers. = Examples. 110. i ole a 221 : n=0 n=0 2" 2 (n+ 1) fn + 2) © mp p\ [2\" a YE Erd.g + ’ ) (5) =37+1. n=0 n=0 § 23. The exponential function. 1. Besides the geometric Seiten; The So colle exponential series Smita Dt Ot n= plays a specially fundamental part in the sequel. We proceed now to examine in more detail the function which it represents. This so-called exponential function we denote provisionally by FE (x). As the series converges everywhere by 92,2, E (x) is certainly, by 98, defined, continuous and differentiable any number of times, for every x. For its derived function, we at once find n! E'(z)=E (x), SO that for all derived functions of higher order we must also have Em) == Ein). ‘We shall attempt to deduce all further properties from the series itself. We have already shown in 91,3 that if x,-and x, are any two real numbers, we have in all cases (a) E (xy + 22) = E (x1) - E (cq) . This fundamental formula is referred to briefly as the addition theorem for the exponential function. It gives further E(x, +a, +x) =E (2, + 2,)- E (x) = E (x) E (x) E (,) and by repetition of this prey we find that for any number of real numbers &,, &,, + ++ Xp» (b) E (x, Sn +2) =E (@,)-E (,)... E(x). 1 Alternative proof. The Taylor's series 99 for E (x) is BE’ E@=E@+ EM a_u)+..., valid for all values of z and z,. If we observe that E® (2) = Z (m,), then it at once follows, replacing = by x, + ,, Co E(t, +) = rEmfi) +5] cE) Ew, ft q.e.4. 192 Chapter VI. The expansions of the so-called elementary functions. If we here write ,—=1 for each », we deduce in particular that Er) =[E@®]" holds for every positive integer k. Since E (0) =1, it also holds for £=0. li we now write; in (b), a, = = for each », denoting by m a second integer > 0, then it follows that 2(e-3)=[e(%)] or, — since E (m)=[E (1)]™, — that E(%) = [EO] If we write for brevity E(1)=E, we have thus shewn that the equation (c) Eg) =k" holds for every rational x > 0. If & is any positive irrational number, then we can in any number of ways form a sequence (z,), of positive rational terms, con- verging to &. For each #n, we have, by the above, Ew)=E™. When 7 — co, the left hand side, by 98,2, tends to E (£), and the right hand side, by 42,1, to E Z so that we obtain E(§) = E" Thus equation (c) is proved for every real 20, But, finally, (a) gives E(—a)-E@=E@—a)=E(0)=1, 2 k . whence we first conclude that E(x) = 0 cannot hold for any real x? and that for # > 0 Blade wt La 20 E (x) EE” But this implies that equation (c) is also valid for every negative veal =, We have thus proved that the equation holds for every real x; and at the same time the function E (x) has justified its designation of exponential function; E (x) is the ath power of a fixed base, namely of 5 5 EE =1+g+ grat tte 2 This may of course, for x => 0, be dcduced immediately from the series, by inspection, since this is a series of positive terms whose term of rank Qis=1. - § 23. The exponential function. 193 2. It will next be required to obtain some further information about this base. We shall show that it is identical with the number e already met in 46,4, so that % 1 3 lim (1 + - += 2 57 The proof may be made in more . comprehensive, by at once establishing the following theorem, and thus completing the investi- gation of 46,4: o Theorem. For every real x, 111. lim (1 +2)" exists and is equal to the sum of the series > it n> 0 Proof. We write for brevity (1+) ==} ond Somme, It then suffices to prove that (s —x,)—0. Now if, — given, first, a _ definite value for x, — ¢ is chosen > 0, we can assume p so large that the remainder LE ern Tomo Sy. Further, for nn > 2, (er xk n\ gts HE + O)R +S —1+adq(1—)e te blo Jf 2 ott a series which terminates of itself at the xh term. The term in x*, k=0,1,..., evidenlly has ‘a coefficient 2 0, but net greater than the coefficient 1/k! of the corresponding. term of the exponential series. The same is also true, therefore, of the difference of the former and the latter term. Accordingly we have, for n > p® — from the manner in which p was chosen — rm i= (=| elt oo ne emp blir 2). (1-229 14 Pde Every individual term of the (p — 1) first terms on the right hand side 3 We have here, therefore, a significant example of problem B. Cf. intro- duction to § 9. 4 First proved — if not in an entirely irreproachable manner — by Euler, Introductio in analysin infinitorum, Lausanne 1748, p. 86. — The exponential series and its sum ¢® were already known to Newfon (1669) and Leibnitz (1676). 5 We assume p > 2 from the first. - 194 Chapter VI. The expansions of the so-called elementary functions. is now obviously the nth term of a null sequence®; hence their sum — for p is a fixed number — also tends to 0, and we may choose fy, > p so large that this sum remains for every w >," But we then have, for every n >, So z,| < & which proves our statement’. — For x = 1, we deduce in particular | Zz-1 : iv E=) —= 1m (1+) =e; v=0 "* y—> © % and more generally, for every real x, B= He The new representation thus obtained for the number ¢, by the “exponential series, is a very much more convenient one for the further discussion of this number. In the first place, we can, by this means, easily obtain a good approximation to ¢. For, since all the terms of the series are positive, we evidently have, for every #, 1 1 lave Ss. e< 8 + a3 + wn ST GEE ! or nt1 §,< eg am nn i.e, (a) S, Ee <8, Ta 8 We have (1-4)~1 (1-2), ing (1-222) 1, and so their n n n a 1 p—1 product (by 41,10), also —»1, or [1 — (1 J an il lee —0; so, as 1 and p are fixed numbers, the product of this last expression by ile)? also — 0; and similarly for the other terms. — We can also infer the result directly from 41, 12. 7? The artifice here adopted is not one imagined ad hoc, but one which is frequently used: The terms of a sequence are represented as a sum 2, = 0" 5 foes oP , where the terms summed not only depend in- dividually on 7, but ali increase in number with n: k, ->00. If we know how each individual term behaves for n-»> 00, as for instance, that x(n) for fixed » tends to &,, then we may often attain our end by separating outa fixed number of terms, say z,™ +, +. ot, with fixed p; this tends, when noo, to §+& +--+ +&p by 41,9. The remaining terms, 20 yb & we then endeavour to estimate in the bulk directly, by finding bounds above and below for them, which often presents no difficulties, provided p was . suitably chosen. § 23. The exponential function. 195 < where s, denotes a partial sum of the new series for e. If we cal- culate these simple values e. g. for n=9 (v. p. 251) then we find 2.718281 < oe < 2-718 282, which already gives us a good idea of the value of the number ¢83. — From the formula (a) we may, however, draw further important inferences. A number is not completely before us unless it is rational ? and is written in the form Is ¢ perhaps a rational number? The inequalities (a) show quite easily that this is unfortunately not the case. For it we had ¢= z then for » = ¢, formula (a) would give: g 1 test where s = 2 ice. +o If we multiply this inequality by g!, then g!ls, is an integer, which we will denote for the moment by g, and it follows that 1 g0, and =, is determined, by 46a, so that | (1 +) e pede n>n, we have y, > n,. remains <{g for every #>mn,, then we shall also have < ¢ for every n> n,, provided #, is so chosen that for every 112. 196 Chapter V1. The expansions of the so-called elementary functions. If the numbers y are not integers, there will still be for each : one (and arly one) ge k, such that ‘ io y. 1, Tr NN, 1 x, 3 [fof bh AYR 1 Lf And since the numbers kare integers, the sequence 1\k,+1 1\%, 1 rE) =r aT rs 5) and the sequence {1+ nr (14 Ee Lo ITTY both tend to e, by our first remark. Hence, by 41, 8, we also have [t+ 3 fe We may next show that when y '— — co, we also have Ti e =e or, otherwise, that when y — + oo, we have (1 Hy 7 e. \ Yn All the numbers V2 must, however, be assumed < — 1, i. e. y,, > 1, so that the base of the power does not reduce to 0 or a negative value; this can always be brought about by “a finite number of alterations”. Since (=) "= Cy) = (5) (1+ 5) and since, with y , y — 1 also — +} co, the statement to be proved is an immediate consequence of the preceding one. Writing =z, we may couple the two results thus: n 1 14-2)n—e provided (z,) is any null sequence with only positive or only negative terms, — the terms in the latter case being all > — 1. From this we finally obtain the theorem, including all the above results: Theorem: If (x,) is an arbitrary null sequence whose terms are different from O and, from the first, > — 1° then 1 (a) lim (14-2, name, 1 nro 10 The latter may always be effected by “a finite number of alterations” (cf. 88, 6). 11 Cauchy: Résumé des legons sur le calcul infinit., Paris 1823, p. 81. § 23. The exponential function. 197 Proof. Since all the z's §= 0, the sequence (x,) may be divided into two sub-sequences, one with only positive and one with only ne- gative terms. Since, for both sub-sequences, the limit in question, as we have proved, exists and = ¢!2, it follows by 41, 5 that the given sequence also converges, with limit e. By 42, 2, the result thus obtained may also be expressed in the form (b) which will frequently be used. log (1+ ,) Ln 1, By § 19, Def. 4, the result also signifies that, invariably: 1 lim(14 x)" =e. ax—>0 From these results, it again follows, — quite independently, as we announced, of our investigations of 1. and 2. —, that pt gr for 2) is certainly a null sequence!3, so that we have, by the pre- ceding theorem, n (x 1 Lad e and therefore (x + 2) —_, — which was what we required. 4. If a> 0, and xz is an arbitrary real number, then 2 oq)? av — ovis —1 | 1082, | (o8aF 0 | (omaly is an expansion in power series of an arbitrary power. We deduce the limiting relation a” OE LL for ax—0 a>0. "113, 12 If one of the two sub-series breaks off after a finite number of terms, then we can, by a finite number of alterations, leave it out of account. 13 We consider this null sequence for » > |x| only, so that we may al- ways have Z>-1. 14 Combining this with the result deduced in 2., that the above limit has the same value as the sum of the exponential series, we have a second proof of the fact that the sum of the exponential series is = e?. 1 Direct proof: If the z,’s form a null sequence, then by 895, 3, so do the numbers y, =a“ — 1; and consequently, by 112 (b), a’ —1 Yn-log a log a = py Ln log (1 4+ v4) 1 = log a. 198 Chapter VI. The expansions of the so-called elementary functions. This formula provides us with a first means of calculating loga- rithms, which is already to a certain extent practicable. For it gives, e.g. (cf 59, p. 76) log a =1limn (Va — 1) n->o 2. =1lm2 Va — 1). k>x As roots whose exponent is a power of 2 can be calculated directly by repeated taking of square roots, we have in this a means (though still a primitive one) for the evaluation of logarithms. 5. We have already noted that ¢” is everywhere continuous and differentiable up to any order, — with ¢* = (¢") = (¢")" =--.. It also shares with the general power 4”, of base a >1, the property of being everywhere positive and monotone increasing with x. More noteworthy than these are the properties expressed BY a a series of simple inequalities, of which we shall make use repeatedly in the sequel, and which are mostly obtained by comparison of the exponential with the geometric series. The proofs we will leave to .the reader. 114. ~ ayforevery uo, & > 11%, DNforw—1, aie, <=, foro t1, zc ming x ¢) for x > —1, Bod {ior 2>0, &>2 (p=0:1, 2, os) zy gyfor x >0and y>0, &> (142) > oo, 9) for every w=:0, |ez—1[< el? —1<|n]elnl, § 24. The trigonometrical functions. We are now in a position to introduce the circular functions rigorously, 1. e. employing purely arithmetical methods. For this pur- pas we consider the series, everywhere convergent by 92, 2: xt 22k Clj=t=G ET bf rm tL 18 Only for x = 0 do these and the following inequalities reduce to equa- lities. — The reader should illustrate the meaning of the inequalities on the relative curves. § 24. The trigonometrical functions. 199 and Bn 2k +1 Sime = Th Gm oi (PE Each of these series represents a function everywhere continuous and differentiable any number of times in succession. The properties of these functions will be established, taking as starting point their ex- pansions in series form, and it will be seen finally that they coincide with the functions cosz and sinz with which we are familiar from elementary studies. 1. We first find, by 98, 3, that their derived functions have the following values: = —3S, C/o C, C"=35, CC" ==(; S'=C, S"—=—8, She on Co SS" —=8; — relations valid for every a (which symbol is for brevity omitted). Since, here, the 4th derived functions are seen to coincide with the original functions, the same series of values repeats itself, in the same order, from that point onwards in the succession of differentiations. Further, we see at once that C(») is an odd, and S(z) an even, function: Shea C(—a)=C@H), S(—2)=—S(. These functions also, like the exponential function, satisfy simple ad dition theorems, by means of which they can then be further examined. They are most easily obtained by Taylor's expansion (cf. p. 191, foot- note 1). This gives, for any two values x, and z,, — since the two series converge everywhere (absolutely), — C (= C ay + 2) = Clo) + Sey + Sar oo, and as this series converges absolutely, we i by 89, 4, rearrange it in any order we please, in particular we may group together all those terms for which the derived functions which they contain have the same value. This gives cr 2a) Clo, +a) = Cla) [1-20 +2 — +. : BONS (a) C(x, + 2,) = C(x,) C(x) — S(x,) S(x,); and we find quite similarly (b) Sx, + x,) = S(2,) C(%,) + C(x) S(z,). ¥* 17 Second proof. By multiplying out and rearranging in series form, we obtain from C (2) C(,) — S (@,) S (wp) the series C(x, + ,), — as in 91, 3 for the exponential series. Third proof. The derived function of f(z) = : [C(x +2) — C(x) C(x) + S (2) S@)E+[S(@, +2) — Sx) C(x) — C(x) S@)] is, as may at once be seen, =0. Consequently (by § 19, theorem 7), f (#)=f(0)=0. Hence each of the square brackets must be separately = 0, which at once gives both the addition theorems. 200 Chapter VI. The expansions of the so-called elementary functions. From these theorems, — whose form coincides with that of the addition theorems, with which we are already acquainted from an ele- mentary standpoint, for the functions cos and sin, — it easily follows that our functions C and S also satisfy all the other so-called purely goniometrical formulae. We note, in particular: From (a), writing ®, = — @,, we deduce that, for every z, (c) C? (@) + S* (2) = 1; from (a) and (b), replacing both x, and x, by x: @ C22) =C) — S*(@) S52a9y=2C) SE). 2. It is a little more troublesome to infer the properties of periodicity directly from the series. This may be done as follows: We have CO=1"0. On the other hand, C (2) < 0; for 92 24 26 98 9210 912 comida = T= where the expressions in brackets are all positive, — since for n > 2, — on 2 n+42 — Sane? —, 4 16 1. : : and therefore C(2)<1— 5 -}- i Ta certainly negative. By § 19, Theorem 4, the function C(x) therefore vanishes at least once between 0 and 2. Since further, as may be again easily verified, s@=c(1—5) +5(1—g%) + is positive for all values of x between O and 2, and therefore C' (x) = — S(x) constantly negative there, — it follows that C(x) is (strictly) monotone decreasing in this interval and can only vanish at one single point & in that interval. The least positive zero of C(x), i. e. &, is accordingly a well-defined real number. We shall imme- diately see that it is equal to a quarter of the perimeter of a circle of radius 1 and we accordingly at once denote it by 5 TT JT t=3, o(f)=e.= From (c), it then follows that S* (z) =1, i.e since S(v) was seen to be positive between 0 and 2, that TT 18 The situation is thus that x is to stand for the moment as a mere ab- breviation for 2&; only subsequently shall we show that this number z has the familiar meaning for the circle. § 24. The trigonometrical functions. 201 The formulae (d) show further that Cla) =—1, Sn) =0, and by a second application, that Coa ="1, S{24)=0. It then finally follows from the addition theorems that, for every =z, Cle+3)=—S@), S(z+3)=C@®, ©) Cleta)=— Cn), SE+a)=—S(), C(x — x)= — C2), S(n—a)=S{)), Cx+2x)=0C(x), S+2x)=8 (x). Our two functions thus possess the period 27.1? : 3. It therefore only remains to show that the number sz, intro- duced by us in a purely arithmetical way, has the familiar geometrical significance for the circle. Thereby we shall have also established the complete identity of our functions C(x) and S(x) with the functions cosz and sinz respectively. Let a point P (fig. 3) of the plane of a rectangular coordinate system OXY, be assumed to move in such a manner that, at the time ¢, its two coordinates are given by z=C{) and" w= SH; then its distance |O P| = Va? + 3° from the origin of coordinates is constantly = 1, by (c). The point P therefore moves along the peri- meter of a circle of radius 1 and centre O. If, in particular, ¢ increases from 0 to 2m, +¥ then the point P starts from the point A of the positive x-axis and describes the peri- meter of the circle exactly once, in the mathe- matically positive (i. e. anticlockwise) sense. “7 In fact, as ¢ increases from Oto, # = C() decreases, as is now evident, from 1 to — 1, monotonely, and the abscissa of P thus assumes each of the values between -- 1 Fig. 3. and — 1, exactly once. At the same time, S (¢) remains constantly positive; this therefore implies that P describes the upper half of the circle from 4 to B steadily, and passes through Zz 2X 1% 2 7 is also a so-called primitive period of our functions, i. e. a period, no (proper) fraction of which is itself a period. For the formulae (e) show that 7 ; ; : oo . =a is certainly not a period. And a fraction =Z, with mm > 2, cannot be a period, as then e.g. S (2) = S (0) =0, which is impossible since S (x) was ~ seen to be positive between 0 and 2 and in fact, as S (zx —2) = S (x), is positive Li between 0 and zz. Similarly for C (xz), Py (m > 1) cannot be a period. 202 Chapter VI. The expansions of the so-called elementary functions. each of its points exactly once. The formulae (e) then show further that when # increases from zz to 27, the lower semi-circle is described in exactly the same way from B to A. These considerations provide us first. with the Theorem. If x and y are any two real numbers for which x®+y*=1, then there exists ome and only ome number t between O (incl.) and 2x (excl.), for which, simultaneously, T C= ani « SQ)=1y- If we next require the length of the path described by P when t has increased from O to a value ¢,, the formula of § 19, Theorem 29 gives at once, for this, the value to 4 [VCP +S a =Jat=y,. In particular, the complete perimeter of the circle is 2x 2x = [VOX S%at— [dt—2a. 0 0 The connection which we had in view between our original conside- rations and the geometry of the circle, is thus completely established: C(t), as abscissa of the point P for which the arc Als ¢, coincides with the cosine of that arc, or of the corresponding angle at the centre, and S(¢), as ordinate of P, coincides with the sine of that angle. From now on we may therefore write cos¢ for C(f) and sin¢ for S(¢). — ~ Our mode of treatment differs from the elementary one chiefly in that the latter introduces the two functions from geometrical considerations, making use naively, as we might say, of measurements of length, angle, arc and area, and from this the expansion of the functions in power series is only reached as the ultimate result. We, on the contrary, started from these series, examined the functions defined by them, and finally established — using a. concept of length elucidated by the in- - tegral calculus — the familiar interpretation in terms of the circle. 4. The functions cotz and tanxz are defined as usual by the ratios cos sin x COLL = —; ” tany = 3 Sin x CoS» as functions, they therefore represent nothing essentially new. The expansions in power series for these functions are however not so simple. A few of the coefficients of the expansions could of course easily be obtained by the process of division described in 109, 4. But this gives us no insight into any relationships. We proceed as follows: In 109, 5, we became acquainted with the expansion 2° 20 The expression on the left hand side is defined in a neighbourhood of 0 exclusive of this point; the right hand side is also defined in such a neigh- bourhood, but inclusive of 0, and moreover is continuous for x= 0. In such case we usually make no special mention of the fact that we define the left hand side for x = 0 by the value of the right hand side at the point. § 24. The trigonometrical functions. 208 > B, B, x? B, x = Fr =1 tp nah e*—1 Vv ve where the Bernoulli's numbers B, are, it is true, not explicitly known, but still are easily obtainable by the very lucid recurrence formula 106. These numbers we may, and accordingly will, in future, regard as entirely known ?!. We have therefore, — for every “sufficiently” small (cf. 103, 2,4) — xr x Foy hae 2 The function on the left hand side is however equal to x x = (ey we?4l Lwelte 2 oN +1 ~ B21 <3 2 @ e2—¢ 2 and from this we see that it is an even function. Bernoulls's nums- bers B,. B,, B, are therefore, by 97,4, all = 0, and we have, using x the exponential series for ¢? and writing for brevity =z SIE hd RC If on the left hand side, we had the signs 4 and — occurring alter- nately, both in the numerator and denominator, we should have pre- cisely the function zcotz. Dividing out on the left hand side by the factor z, so that only even powers of z occur, may we then deduce straight away that the relation 2 2 1 == + 2 4 Zeotg— —on =1 — 2 3 It — oe obtained from our equality by alternating the signs throughout, is also valid? Clearly we may. For if, to take the general case, we have for every sufficiently small 2: 14a, +a,z2t}. Pe ree Be Get ih the same relation holds good when the -- signs throughout are re- placed by alternate 4 and — signs. In either case, in fact, the coeffi- cients ¢,, are obtained, according to 1035, 4, from the equations: &+b,=a,; ¢,tc,b,+b,=a,; ¢+¢b,+c,b +b =a; Coy + Cay_2by, +--+, bay—2 + 02, = aa,; —-14 5 Er ept... Z LRP + (2 — ++ in eee S #1 As appears from the definition, they are certainly all rational. 116. 204 Chapter VI. The expansions of the so-called elementary functions. We therefore, as presumed, — now writing « for z, -— have the formula: 28, ot 2k p 2 x2k xcoty=1——— i dels + 4 (—1F Ee Banos? LE 101 2 5; 1 8 Bs TE? tam om” ) The expansion for tan x is now most simply obtained by means of the addition theorem SraighauSTmannty Ly — tanz, cos x-sinx from which we deduce tanx = cotx — 2cot2x and therefore 02% 92% _1y B. (a) tangp= 3 =n ped ) Bor yore = en! lL ead 17 or 23 3 15 gs FT ? From the two expansions, with the help of the formula z 1 cotx -- tal 5 == we obtain further 0 » @*-9)B,, = NE TEE (b) sing? = 1) CK)! =i 57 + get + Toa ® 95 of os (An expansion for 1/cosz will be found on p. 239.) — These ex- ‘pansions, at the present point, are still unsatisfactory, as their interval of validity cannot be assigned; we only know that the series have a positive radius of convergence, not, however, what its value is. 5. From another quite different starting point, Euler obtained an interesting expansion for the cotangent which we proceed to deduce, especially as it is of great importance for many problems in series ?% At the same time, it will give us the radius of convergence. of the series 1135 and 116 (v. 241). 22 This and the following expansions are almost all due to Euler and are found in the 9t and 10% chapters of his Introductio in analysin tnfinitorum, Lausanne 1748. 2 We shall afterwards see that By, has the s'gn (-1/%-1 (v. 136), so that the expansion of @ cota, after the initial term 1, has only negative coeffi- cients, those of tanz and Si only positive coefficients. 24 The following considerable simplification of Euler's method for obtaining the expansion is due to Schriter (Ableitung der Partialbruch- und Produkt- entwicklungen fiir die trigonometrischen Funktionen. Zeitschrift fiir Math. u. Phys., Vol. 13, p. 254. 1868). § 24. The trigonometrical functions. 205 We have, as was just shewn, — = y 2) Coby = cols — ian or i a 7x41) *) ota Fle + cot Zen 3 a formula in which we may, on the right, take either of the signs +. Let x be an arbitrary veal number distinct from 0, +1, + 2, whose value will remain fixed in what follows. Then aacotar — 7 leot ZF + cot ZED and applying the formula (*) once more to both functions on the right hand side, taking for the first the } and for the second, the — sign, we obtain ma cotas =F {co afl om? ZLil al pels duel A third similar step gives, for mx cotzx, the — + cot gen + cot Zhe + cot ed Tr = cot 22 ® + cot zi 39 * peat FD Lon Zll t on Zid) >of since here each pair of terms which occupy symmetrical positions relatively to the centre (@) of the aggregate in the curly brackets give, except for a factor 7, a term of the preceding aggregate, in accordance with the formula *). If we proceed thus through # stages, we obtain for n>1 oft =1_4 = = Lh macotaw =F {cot TE + = amt & od cot "GT ran 32 Now by 115, limzcotz=1 z—>0 and hence for each «== 0 oq Im cot r= v3 Si on 2 0’ if in the above expression we let n— co and, at first tentatively, carry out the limiting process for each term separately, we obtain the ex- pansion nacotar—1+2 3 (3 We proceed to show that this in general faulty mode ha ja to the limit has, however, led in this case to a right result. We first note that the series converges absolutely for every x= +1, +2, ..., by 70, 4, since the absolute values of its terms . 3. gi % iy — 92 z+» 2006 Chapter VI. The expansions of the so-called elementary functions. are asymptotically equal to those of the series xl Now choose an arbitrary integer k > 6 |x|, to be kept provisionally fixed. If # is so large that the number 27-1 — 1, which we will denote for short by m, is > k, we then split up the expression (}) for ma cot, as follows: k m ma cota w= Gr cot 7 = tn iS 5 S11 (In the square brackets we have of course to insert the same expression as occurs in (1).) The two parts of this expression we denote by 4, and B_. Since 4, consists of a finite number of terms, the passage to the limit term by term is certainly allowed there, by 41,9, and we have E imd =1-1222Y iS n—>w y=1% Bb Also B, is precisely mx cotnx — A,, hence lim B, certainly exists. Let 7, denote its value, depending as it does upon the chosen value £; thus lim B, =7,=naxcotmnx — [1-20 Si ES n> xo r=1 Bounds above for the numbers B,, hence for their limit », and so finally for the difference on the right hand side, may now quite easily be estimated: We have = —9 cota cot (a + b) + cot (a = b) mm Peay sina and hence a (x +») ot2l=1 —2cota cot on -+- co on =r sin? 8 ’ sin? writing for the moment = « and gr ==}, for short. As 2" > Ek > 6|x|, we certainly have |a| = 3 ; 3 | a sine] =e — 5+ | lal (+ G+ 5+) <2le|™ Since, further, SRfg tn we have sinf} — p(1- etl Situs tiy 2 Cf. Footnote 7, p. 194. 2 For the sake of later applications we make these estimates in the above rough form. 27 Cf. p. 200. ry : § 24. The trigonometrical functions, 207 sin 8 8 sine|” 6]e] = Hence >1, the latter, because y > k > 6 | 2 | . It therefore follows that (for » > k) ( +9) 2 cot 3 [eZ] Loe + co Ea) = 22 36 22 1 and hence TT wx 2. 723 B.S | meote| gem The factor outside the sign of summation is — quite roughly estimated — certainly < 3; for was <1, and for <1 we have on zi zt intl tp Late ! 26 jzcotnl = a 4s 222 yoxl 1 nr TT Accordingly, : AL WEE TES BERR TE v=k+1 36 2* seal LT 6a But this is a number quite independent of #, so that we may also write : . HZ 1 impB. [s=17 1x 216% . 2 ETE But the bound above which we have thus obtained for 7, is equal to the remainder, after the kth term, of a convergent series 2%. Given ¢=0, we can therefore choose k, so large that, for every k > k,, we have ln] » or, as asserted, x eoinpelt It 3 ry 117. — a formula which is thus proved valid for every x = 0, +1, 4 2, 6. We shall in the very next chapter make important applications of this most remarkable expansion in partial fractions, as it is called, of the function cot. We can of course easily deduce many further such expressions from it; we make note of the following: 28 The a is obtained just as simply as, previously, that of the series 2 208 Chapter VI. The expansions of the so-called elementary functions. The formula 7 cot 5 —2acotnz =mtan ZT first of all gives x ® 4x atan Tl = => we tl, £9, 48,0 d = @7 41)? —a? = 1 1 ee: rrr et al The formula z 1 cotz tan 5 = then gives further, for x Fo +2, 2 x R= pty tye tl _— T ls i a Toa Finally if we here replace # by 1 — xz, we deduce 7 2 2 2 ) ( 2 cosmx EE a 8-29 Z\31%g 5 = By 83, 2, Supplementary theorem, the brackets may here be omitted. But if we then take the terms together again in pairs, starting from the beginning, we obtain, — provided x4 +1, +3, +3, ..., b/ 1 4-1 4.3 4.5 118. le 2 = > —— Fe COSADL Pid? Bt? Pda With these expansions in partial fractions for the functions cot, tan, 1 il 2 5 : . : = and Sn we will terminate our discussion of the trigonometrical mn functions. § 25. The binomial series. We have already, in § 22, seen that the binomial theorem for positive integral exponents, if written in the form 2 (5 a+ar=5(,)e n=0 remains unaltered in the case of a negative integral 23°. But we have then to stipulate |2| <1. We will now show that with this restriction 29 The formula first follows only for x40, +1, +2, ... but can then be verified without any difficulty for t=0, +2, +4, .... (The series has the sum (, as is most easily seen from the second expression, for an even integral x.) % In the former case the series is infinite only in form, in the latter it is actually so. 3 § 25. The binomial series. 209 the theorem holds even for any real exponent «, i. e. Oops 32 ae HIRT — « any real number 31, As in the preceding cases, we will start from the series and shew that it represents the function in question. The convergence of the series for | # | < 1 may be at once established; for the absolute value of the ratio of the (n 4 1)th to the nth term is «—n rar which by 76, 2 proves that the exact radius of convergence of the binomial series is 1. It is not quite so easy to see that its sum is equal to the — of course positive — value of (1 z)*- If we denote provisionally by f, (x) the function represented by the series for je] <<, the proof may be carried out as follows. and therefore —» tel, : Since = ) a" converges absolutely for |#| < 1, whatever may be the value of «, it follows, by 91, Rem. 1, that for any « and fp, and every |z| <1, we have S(e0e 80 = ZH + (ILE) + (EN n=0 n=0 n=0 EOL) 020 (H(E)+(; ap 14 o/ n as may quite easily be verified e. g. by induction 32. Hence — for 31 The symbol (2) is defined for an arbitrary real « and integral n >0 4 by the two conventions [E)=ts (O-Ring: and for every real « and every n> 1, it satisfies the relation, which may at once be verified by calculation: nH n—1 n An/ 3 For this, give the statement, by multiplying by #!, the form (5) FG=D Fmt D+ +(3) FED EFI D-FE-D...F=n FEF D+... +(3)e@ DE nT D=( +h (+=... +f—n+1). Then multiply each of the (n 4 1) terms on the left hand side first by the corres- ponding term of : a u—=1),...,{¢a—8)y ..., (=n), then by the corresponding term of B—mn), B—n+1), oe ty (B—n+E), ve sy pg and add, so that in all we multiply by (e+ B — mn); grouping together the similar terms on the left, we obtain precisely the asserted equality, where # is replaced by n+ 1. — The above formula is usually called the addition theorem for the binomial coefficients. > 119. 910 Chapter VI. The expansions of the so-called elementary functions. fixed |¢| <1, — we have, for any « and 8, fe y 13 = fats . By precisely the same method as we used to deduce from the addition theorem of the exponential function, — E (z,)-E (z,) = E (2, + x,), — that for every real z we had (E (x)= (E (1), — so we could here conclude that for every «, * fa = i) 5 if we knew here also that f, was for every real « (with fixed 2) a continuous function of a. As f, =1-z, the equality f a == (1 + z)* would then be established generally for the stated values of x. The proof of the continuity results quite simply from the main rearrangement theorem 90: If we write the series for f, in the more explicit form o? o © fittest er(I-Srf)ete and then replace each term by its absolute value, we obtain the series ® lal(le] +]: (al+n=1) g(lel+n-1 Ean RT AR also convergent for |2| <1 by the ratio test. We may accordingly rearrange the above series (a) in powers of «, obtaining O fmt rT ET Jer i. e. certainly a power series in «. Since this — still for fixed = in || <1 — converges, by the manner in which it was obtained, for every oe, we have an everywhere convergent power series in «, hence certainly a continuous function of e. This completes the proof of the validity of the expansion 1193 and at the same time fills the gap left in the proof of the reversion theorem § 21. 38 Ap alternative proof, perhaps still easier than the above, but using the differential calculus, is as follows: From f, (x) = >] 5 am it follows that : n=0 fr0=Sa(Ret= Zo +0(, 4) ) =a {* 3 , it follows further that n+ 1 n fo @=cfo_y@)- Since however (r41) ( But A+f, @=0+a)- F (5, Der 0 n= jor cine IYI Lee 2 Che BE tft § 26. The logarithmic series. 211 The binomial series provides, like the exponential series, an expansion of the general power a¢: Choose a (positive) number ¢ for which, on the one hand, ce may be regarded as known, and on the other, 0<=-<2. Then we may write S=1+= with |#| <1 and so ob- tain, as the required expansion, ae = eo (1 +a) —ce [1+ (Hat (a2 +]. Thus e. 2. vi-o= (8) = fi —5l1- (Da+ Fm (Des + = is a convenient expansion of V2. The discovery of the binomial series by Newton®* forms one of the landmarks in the development of mathematical science. It was suggested by J. Wallis’s investigations on interpolation: Newton gave no proof; James Bernoulli published a proof for positive integral in- dices in 1713, and Euler gave an incomplete proof of the general theorem in 1774. Later Abel ®® made this series the subject of re- searches which represent a perhaps equally important landmark in the development of the theory of series (cf. below 170, 1 and 247). § 26. The logarithmic series. As already observed on p. 56, in theoretical investigations it is convenient to employ exclusively the so-called natural logarithms, that is to say, those with the base e. In the sequel, loga will therefore always stand for log, x (x > 0). i y=1logz, then a==0¢7 or 02 s—1=y+ S84. By the theorem for the reversion of power series (107), y=logx is thus, for every || <1, we have the equation (+2) fe@—eaf,(@=0 Since (1 +2)* > 0, this shows that the quotient fa® (1+ 2)* has everywhere the differential coefficient 0, i. e. is identically equal to one and the .same constant. For x= ( the value is at once calculated and = + 1; thus the assertion f, (x) = (1 + 2)* is proved afresh. 3 Letter to Oldenburg, 13 June 1676. — Newton, however, was in possession of the formula some years earlier. 8 J. f. d. reine u. angew. Math,, Vol. 1, p. 311, 1826. 120. 212 Chapter VI. The expansions of the so-called elementary functions. therefore expansible in powers of @ — 1) for all values of z sufficiently near to 41, or y =log(1 + 2) in powers of z, for every sufficiently small |z |: y=Ilog(1+2)=a+b,22 by a®4----. It would no doubt be quite useless to attempt to evaluate the general coefficient b, by the method indicated. We have to content ourselves: with using the reversion theorem to prove the possibility of the ex- pansion. To determine the coefficients b,, we must seek more con- venient methods: For this purpose, the developments of the preceding section suffice. For || <1 and arbitrary «, the function f, =f, (2) there examined is zed 1 1 2)" = galog(1+2), Using, for the left hand side, the expression (b) of the former paragraph and for the right hand side, the exponential series, we obtain the two power series everywhere convergent: tt Er Tet =1+ [log(1+4 2)]« By the identity theorem for power series 97, the coefficients of corre- sponding powers of ¢ must here coincide. Thus, in particular, — and for every |z| <1 — (2) tog (14 0) = — 5 + Fr — + AE eng. Thus we have obtained the desired expansion, rh we also see a posteriori, cannot hold for |x| > 1. If we replace in this logarithmic series, as it is called, x by — x and change the signs on both sides of the equality, we obtain, — Sly. for every 1 <1, (5) pepe =t+ OT Tf By addition we deduce, — pts for every |z| 1, Es 22k+1 > © tgltfeaat THT TN] There are of course various other ways of obtaining these expansions; but they either do not follow so immediately from the definition of the log as inverse function of the exponential function, or make more extensive use of the differential and integral calculus?” ey peer 38 Cf. the historical remarks in 69, 8. 37 We may indicate the following two ways: 1. We know from the reversion theorem that we may write log(l4+a)=x+bya®+bya3 4; it follows from Taylor's series 99 that b 1 /d*log (1+ 2) fDi = 7 dx* ), =0 - § 27. The cyclometrical functions. 213 Our mode of obtaining the logarithmic series — also the two modes mentioned in the footnote — do not enable us to determine whether the representation remains valid for x= 41 or x = — 1. Since however 120a reduces, for x = -}- 1, to the convergent series (v. Ste, 3) ' 1 1 il i=—t-by=g Te, the value of this series, by Abel's theorem of limits, is = lim log(14 a)=1log 2. Sealab Our representation (a) there remains valid for x = -}- 1; but for # = —1 it certainly no longer holds, as the series is then divergent. § 27. The cyclometrical functions. Since the trigonometrical functions sin and tan are expansible in power series in which the first power of the variable has the coeffi- cient 1, different from 0, this is also true of their inverses, the so- called cyclometrical functions sin~! and tan™!. We have therefore to write, for every sufficiently small |x|, y=sinle=x-b,2® tb, a® 4... y=tan lz =x +b 2% +b, a* J... where we have out the even powers at once, since our functions are odd. Here too it would be futile to seek to evaluate the coefficients b and b’ by the general process of 107. We again choose more con- venient methods: The series for tan”! is the inverse of 5 shy ye Bt a (a) = w= Beef Yims Soi sain re 2 i or of the series obtained by 109, 4 alter carrying out the process of division in the last quotient. If here all the signs, in numerator and denominator, were -, then we should be concerned with reversing the function el=e=Y 2.1 no = a . EE em sn | ee Foret (Dik. = ih Integrating, it follows at once, by 99, theorem 5, — since log ¥ =(0, — that hi oh tog (1-40) = 3 7 is =e > The method in the text is so 5 simpler at it ha entirely without : the use of the differential and integral calculus. 8% 121. 122. 214 Chapter VI. The expansions of the so-called elementary functions. But the inverse of this function is, as we immediately find, 1+ zs 20. =zt=t 54. 1—2 1 9 log By the general remark at the end of § 21, the reverse series of the series for @ = tany actually before us is obtained from the series last written down by alternating the signs again, i. e. a? xr tan lop=0——4+——:.-. 38 If therefore this power series, which obviously has the radius of con- vergence 1, is substituted for y in the quotient on the right of (a), and this is then rearranged, as is certainly allowed, — we obtain =z. Hence its sum for an arbitrary given |2| <1 is a solution of tany =z, and is precisely the so-called principal value of the function tan=1z hereby defined, i. e. the value which is = 0 for # = 0 and then varies continuously with x. This value lies, for —11 the expansion obtained is certainly no longer valid; but Abel's theorem of limits shews that it does still hold for x = + 1. For the series remains convergent at both end-points of the interval of convergence and tan~!z is continuous at both these points. We have therefore in particular the series, peculiarly remarkable for clearness and simplicity: x 1 x 1 iT rits it giving at the same time a first means of determining zz of some prac- tical value. This beautiful equation is usually named after Leibni£23?; it may be said to reduce the treatment of the number mz to pure arithmetic. It is as if, by this expansion, the veil which hung over that strange number had been drawn aside. 8 A different method is the following: We have dtan— 1g 1 1 1 ] ; je IIE TITY Jat —+F..., or the latter for |z| <1. As tan=10=0, it follows by 99, theorem 5, that for lz| <1, at go =1 == —_—— em ces, tan— tx =p gtr + — A method corresponding to that given first in the preceding footnote is some- what more troublesome here, as the differential coefficients of higher order of tan—!2 — even at the single point {} — are not easy to find directly. — The expansion of tan—!x was found in 1§71 by J. Gregory, but did not become known till 1712. : 39 He probably discovered it in 1673. Exercises on Chapter VI. 215 For the deduction of a series for sin"1x, the method which we have just used for tan~'x is not available. The process indicated in the last footnote, however, provides the desired series: We have for |z]| <1 dsin='=zx 1 I 1 n—3 dz /dsiny) Gv ft 2 5 ay ig the positive sign being given to the radical since the derived function of sin"'x is constantly positive in the interval — 1...4 1. From == =3 2 =1 4 (sin Wf =1~ (Fla +(3)et = + it at once follows, however, by 99, theorem 5, as sin~10 == 0, that for || <1 1:3 »® , 1.35 =’ -_1 —— ese gi w= 412. Te Tr 123. This power series also has radius 1, and on quite similar grounds to the above we conclude that for || <1 its sum is the principal value of sin"!x, i. e. that uniquely determined solution y of the equation siny = x which lies between — 5 and +3 For x = + 1, the equality is not yet secured. By Abel's theorem of limits it will hold there if, and only if, the series converges there. As we have a mere change of sign in passing from + 2 to — z, this only needs testing for the point 4 1. There we have a series of po- sitive terms and it suffices to show that its partial sums are bounded. Now for 0 < «<1, if we denote by s, (x) the partial sums of 123, s,) —~=1g", with the same meaning of %, as in d). 2 1-2! * =» 75. Show that the expansions in power series of the following functions begin with the terms indicated: x phi? a a) re lm m ayy eens 3 log l1—=2 y aM pm b) (1—2a)e wi mt m=1); c) tan (sin x) — sin (tan Hg eine 9 ; Nem” tam x : ate talon Luma 2 0 9 niin § 5 1+ atm? To” pL x? 4 = ep? 3 Seno AE 3st 76. Deduce, with reference to 103, 5, 115 and 116, the expansions in power series of the following functions a) log cos x; b) log 02; Sr x log 3 sin x x2 1 9) 1— cosa’ f cosa’ xz : 1 g) logy bh) ad 2sin — a 2 NE 1 ) ry = cosx—sinz $7. Show that 1 ft tT 21.3 2° . FE eit tl el a1 (a+1)(a+3) 2 1. = [1+ PT e+e dh” x ¥S. We have 2 : =) —e. Is the sequence monotone? Increasing or decreasing? What, in this respect, is the behaviour of the sequences + (145) Oxoal? 79. From x, — £ it invariably follows that es | Exercises on Chapter VI, 217 and also, if x, and & are positive, that n(Yzs—1)—>logk. ah 80. If (z,) is an arbitrary real sequence, for which ~*~ — 0, and we write Ta\" (1 - 2) =4y,, then, in every case, n Vo DE "a, 81. Prove the inequalities of 114. 82. Express the sums of the following series by closed expressions in terms of the elementary functions: 1 oo op? gf Betstriu (Hint: If f(x) be the required function, then obviously @f 0 = 1 1—23 whence f(z) may be determined. Similarly in the following examples.) 23 2 x2 Bw rsTg md 1 xz x2 rhs rriT rast x zs EE aaa 83. Obtain the sums of the following series as particular values of ele- mentary functions: 3 4 St mtapt gy tee=l r 1.3 133.5 Tigh tar trons unl 1-3 1:3:5.7 1.3.5.7.9:11 1 V3t54672.4.681072.46810-12.14 BVH ¥ 13 1-8:5.7 1-3:5.7.9-11 Te-1 al i - VE 8 97 6'3.1.¢.070 510010-13-1% V2 84. Deduce from the expansion in partial fractions 117 seq. the following expressions for sz: x 1 1 1 1 ] spent reds tr mint te 1 1 1 1 ] m= oil. =i tot where ¢ 4-0, 4+ 1, + 4, + 4,... Substitute in particular «=3, 4, 6 J. 218 Chapter VII. Infinite products. Chapter VIL Infinite products. § 28. Products with positive terms. An infinite product Uy Uy Wyo Uh, > is, by § 11,11, to be taken merely as representing a new symbol for the sequence of the partial products > Su Pn=Uy Uy... U,. Accordingly such an infinite product should be called convergent, with value U, 0 Hu, =U, n=l if the sequence (p,) tends to the number U as limit. But this is parti- cularly inconvenient, owing to the fact that then every product would have to be called convergent for which a single factor was = 0. For ff u,=0, then p,=0 for every n =m, and hence p,—U=0. Similarly every product would be convergent — again with the value 0 — for which from some mm onwards | u, | = 9 m — no factor vanishes, and if the partial products, beginning immediately beyond this point 3 = Yniy Ym’ Uy (n> m) tend, as m increases, to a limit, finite and different from 0. If this be = U,,, then the number U=wu,-u,....u,-U,,, obviously independent of m, is regarded as the value of the product’. m’ 1 Infinite products are first found in F. Vieta (Opera, Leyden 1646, p.400) who gives the product 2 VLV LIVE ViVi IE § 28. Products with positive terms. 219 We then have first, as for finite products, the oTheorem 1. A convergent infinite product has the value 0 if, and only if, one of its factors is =O. As' further p_ _,—U, with p,—U,, and as U, is =0, we have (by 41,11) and we have the o Theorem 2. The sequence of the factors in a convergent infinite product always tends — 1. On this account, it will be more convenient to denote the factors by u,=1+a,, so that the products considered have the form : (+a). For these, the condition a,— 0 is then a necessary condition for con- vergence. The numbers ¢, — as the most essential parts of the factors — will be called the terms of the product. If they are all > 0, then as in the case of infinite series, we speak of products with positive terms. We will first concern ourselves with these. The question of convergence is entirely answered here by the Theorem 8. A product II (1--a,) with positive terms a, is convergent if, and only if, the series > a, comverges. Proof. The partial products p,=(1-44,)...(1+a,), since a, > 0, increase monotonely; hence the First main criterion (46) is available and we only have to show that the partial products p are bounded if, and only if, the partial sums s, =a, +a, +--+ a, are bounded. Now by 114 « 1--a, < ¢% and so-for each n Ins on the other hand p.=0+a)...04a)= 14a, + a, Bfriee +a, a, a = Sys the latter because in the product, after expansion, we have, besides the terms of s , many others, but all non-negative ones, occurring. Thus for each # 50, (cf Ex. 89) and in J. Wallis (Opera I, Oxforl 1695, p.468) who in 1656 gives the product But infinite products first secured a footing in mathematics through Euler, who . established a number of important expansions in infinite product form. The first criteria of convergence are due to Cauchy. 220 Chapter VIL Infinite products. The former inequality shows that $_ remains bounded when s, does, the latter, conversely, that s, remains bounded when p, does, — which proves the statement.? / Examples. 1. As we are already acquainted with a number of examples of con- vergent series Xa, with positive terms, we may obtain, by theorem 3, as many 4 examples of convergent products II (1 +a,). We may mention: II (1 +) is convergent for « >> 1, divergent for « < 1.— The latter n is more easily recognised here than in the corresponding series, for 1 1 IV 2.8 ¢ ail (1+) (14g) (14) m1 3 5 =n +1to0. 3 2. 74 +m) is SonySIgen for 0 < x <1; similarly IT(1+ x2"). e-D@E+2_1 JI (1 Tren ese a With theorem 3 we may at once couple the following very similar Theorem 4. If, for every mn, a, > 0, then the product II1(1 — a) 1s also convergent if, and only if, > a, converges. Proof. If a, does not tend to 0, both the series and the pro- duct certainly diverge. But if 4, — 0, then from some point onwards, say for every n >m, we have a, <1, or 1—a >1. We consider the series and product from this point onwards only. Now if the product converges, then the monotone decreasing sequence of its partial products p =(1—a,,,)..-(1 —a,) tends to a positive (> 0) number U,, and, for every n > m, l—a,.q)---Q—a)=2U,>0. Since, for 0 < a, << 1, we always have (+e) <1 (as is at once seen byf@multiplying up), we certainly have (1 @ppty) (1 1 pia) eo a - a,) = Tr 2 In the first part of the proof of this elementary theorem, we use the transcendental exponential function. We can avoid this as follows: If Xa, =s converges, choose m so that for every n> m 1 brit mist TH <7 As, obviously, for these #'s, we now have (A+api)---QA+a) ES 1+ @nyr+ Fa) +@uyr +e Fa)+--- + @m +1 iv oe oa)” == 2, we certainly have, for all #'s, : Pa5>0; and the partial products on the left hand side, as they form a mono- tone decreasing sequence, therefore tend to a positive limit: i. e. the product II(1 — a,) is convergent. Remarks and Examples. 1 0 1 J] (1 -3) is convergent for a > 1, divergent for « <1. n=2 2. If a, <1 and if Xa, diverges, then IT(1 —a,) is not convergent, with our definition. As however the partial products p, decrease monotonely and remain > (, they have a limit, but one which is necessarily = 0. We say that the product diverges to 0. The exceptional part played by the number 0 thus involves us in some slight incongruity of expression. A product is called divergent whose partial products form a decidedly convergent sequence, namely a null sequence, (p,). The addition “in the stricter sense” to the word “con- _ vergent” in Def. 125 is intended when necessary to serve as a reminder of this fact. 8. That e. g. J] (1 -) diverges to 0 is again very easily seen from n=2 ta=(1-) (1-5) (1-7) = 2 a t= 0, 9 3 nj 28 n § 29. Products with arbitrary terms. Absolute convergence. If the terms a, of a product have arbitrary signs, then the following theorem — corresponding to the second principal criterion 81 for series — holds: : oTheorem 5. The infinite product Il (1+ a,) converges if, and 4 A full and systematic account of the theory of convergence of infinite products may be found in A. Pringsheim: Uber die Konvergenz unendlicher Produkte, Math. Annalen, Vol. 33, p. 119—154, 1889. 127. 299 : Chapter VII. Infinite products. only if, given e > 0, we can determine ny So that for every mn > ng and every k > 1, : [a oF 2,41) (1 1 @y1s)- ? (1 = Bir) = 1] m, we have a,== — 1, and the partial products - po=0+a,sy)..-(1+ a), (n > m) tend to a limit <= 0. Hence there exists (v.41, 3) a positive number f such that, for every n >m, |p,| => >0. By the second principal criterion 49 we may now, given ¢> 0, determine n, so that for every n > n, and every k>1, Fa < 2-f. But then, for the same # and Z&, Drpk yy Pn ={{1t1v JT+a,,)...0+2,.)=1]<4 which is precisely what we asserted. b) Conversely, if the e-condition of the theorem is fulfilled, first 1 choose ¢ = 3, and determine m so that, for every un > m, Ata, 0 te)=1l=[p —1] <3 For these n’s we then have 3 m, we must have 1a, ==0; and further, that if p, tends to a limit at all, this certainly cannot be 0. But we may now, given & > 0, choose the number n, so that for every n> mn, and every k >1, Pn +k 3 WE 1] = or [Bair — ul <|Pa]-5 1; or — v. 81, 3rd form — if invariably [A+ arp). (AF Grp] > 1. / § 29. Products with arbitrary terms. Absolute convergence. 298 This definition only gains significance through the theorem: oTheorem 6. The convergence of II(1-|a,|) involves that of II1+a,). Proof. We have invariably A+ a) +a) 4 + tr) 1] | oblige DL basalt =hilpasl) = as is at once verified by multiplying out. If thercfore the necessary and sufficient condition for the convergence of Theorem 5 is satisfied by II(1+|a,|), it is ipso facto satisfied by II(1--a,), q.e.d. In consequence of Theorem 3, we may therefore at once state oTheorem 7. A product II(1-a,) is absolutely convergent if, and only if, >a, converges absolutely. As we have an already sufficiently developed theory for the determination of the absolute convergence of a series, Theorem 7 solves the problem of convergence in a satisfactory manner for absolutely convergent products. In all other cases, the following theorem reduces the problem of convergence of products completely to the correspond ing one for series: ~ oTheorem 8. The product II(1 + a,) converges if, and only if the series 3 log (1+ a,) n=m+1 ’ commencing with a suitable index®, converges. And the convergence of the product is absolute if, and only if, that of the series is so. ~ Furthermore, if L is the sum of the series, then Ha+a)=@+a). (14a) el. Proof. a) It na + a,) converges, then 4, — 0 and hence from some point onwards, say for every un > m, we have | 2,1. 1. Since, further, the partial products po=01"4ap4y..-(11a,), (n> m), tend to a limit U, 3= 0 (hence positive), we have by (42, 2), log p,—1logU,,. ~ But log p, is the partial sum, ending with the nth term, of the series in question. This, therefore, converges to the sum L = log Us As U_=e¢?, we thus have Hl+a)=(0144a,)...0+ a) en b) If, conversely, the series is known to converge, and to have the sum L, then we have precisely logp, — L, and consequently _ (by 42,1) D.. = log Pn» gL, This completes the proof of the first part of the theorem, since eZ==0. It suffices to choose m so that for every n> m we have fay <1. 2924 : Chapter VII, Infinite products. To deduce, finally, that the series and product are, in every possible case, either both or neither absolutely convergent, we use with theorem 7, the fact that (112, b), when 4, —0 log (1+ a,) An —1. (Here any terms ag, which = 0 may be simply omitted from con- sideration.) Although we have thus completely reduced the problem of the convergence of infinite products to that of infinite series, yet the result cannot entirely satisfy us, because of the difficulties usually involved in the practical determination of the convergence of a series of the form log (1-4). The want here felt may, at least partially, be supplied by the following ©Theorem 9. The series (starting with a suitable initial index) 2log (1 + a,) and with it the product I1(1 +- a,), is certainly convergent, if 2a, converges and if Xa? is absolutely convergent. Proof. We choose m so that for every n > m, we have |a,] <1, and consider II(1 + a,) and 2'log (1 | a,), starting with the (m | 1)th terms. If we write log(1+4a,)=a,+9,-a, or bgldte)—a 8 P= 5 2.5 then the numbers ¢, so determined certainly form a bounded sequence, for, as 4,—0, #,— — 1% Ii therefore 2g, and X|a |* are con- vergent, log (1+ a), and hence also IT(1-} a), is convergent. This simple theorem leads easily to the following further theorem: © Theorem 10. If Za? is absolutely convergent, and |a,| is <1 for every nm > m, then the partial products n n p,= I 1+ a) and the partial sums s,= > a,, (n > m), r=m+1 r=m+1 are so related that Pn ~ €" 1. e. the ratio of the two sides of this relation tends to a definite limit, finite and == 0, — whether or no Xa, converges. ? ¥a,2 if convergent at all, is certainly absolutely convergent. We adopt the above wording so that the theorem may remain true for complex as, for which a,? is not necessarily > 0 (cf. § 57). 8 Por 0< |2] <1 we have in fact 1 ag - pli oo al log (1+) =z +2* | 5 +3 rr ] or log (1+2)—= x2 1 z = eck ibe And those terms which are possibly =(0 may be again simply neglected, as they have no influence on the question under consideration. ERTL + RAO g 29. Products with arbitrary terms. Absolute convergence. 295 Proof. If we adopt the notation of the preceding proof, then, as log(1-}a,)=a,+%,a,, we have for every n> m 14 apd h a) = J] St Fe Et, v=m+1 » if the sums in the last two exponents are taken also from » =m }- 1 to vy =mn. And as X¢, a2, the 9's being bounded, converges absolutely when Xg ? does so, we can, from the above equation, at once infer the result stated. — This theorem also provides the following, often useful Supplementary theorem. If Xa? converges absolutely, then Xa, and II (1+ a,) converge and diverge together. Remarks and examples, 128. 1. The conditions of Theorem 9 are only sufficient; the product II (1+ a,) may converge, without Xa, converging. But in that case, by Theorem 10, 2a, |? must also diverge. 92. If we apply theorem 10 to the (divergent) product 7 (1 pd, then it follows that en ~ #" 1 1 if %, denotes the nth partial sum of the harmonic series 4, =1 4 > Heat h « Accordingly the limits lim tL =c¢ and lim[h,—logn]=1logc=C exist, the latter because ¢ #= 0, hence > 0. The number C defined by the second limit is called Euler's or Maschevoni’s constant. Its numerical value is C = 0-5772156649 . (cf. Exercise 86a and 176, 1). The latter result gives us further valuable in- formation as to the degree of divergence of the harmonic series, as it gives h,~logn, Further the estimates of bounds above made for the proot of Theorem 3 show, # : ; 1 even more precisely, if we there put = s that dn>dn-1>n or hy > h, 4 >logn so that Euler's constant cannot be negative. €D (— r= 1 (1 5] found at once by forming the partial products, and is =1. ) is convergent. Its value riay, as it happens, be @0 4. Ble) diverges for x 40. However, theorem 10 shows that n=1 1 ® fel tees tien 7 (+3) ~ 2 Tt +3) or — what is the same thing by 2, — ~ 2%, i. e. (v. 40, def. 2) the ratio n was fey CEB) ott) r=1 n! n® 296 - Chapter VIL. Infinite products. : has, for every (fixed) #, when n— 00, a determinate (finite) limit which is also different from 0 if x is taken f= —1, —2, ... (cf. below, 219, 4). le) 2 5:11 (1 - is absolutely convergent for every x. n=1 " 2 1 1 6. (1 — 5 =. 2 n? 2 § 30. Connection between series and products. Conditional and unconditional convergence. We have more than once observed that an infinite series X'q_ is merely another symbol for the sequence (s, ) of its partial sums. Apart from the fact that we have to take into account the exceptional part played by the value 0 in multiplication, the corresponding remark holds good for infinite products. It follows that, with this reservation, every series may be written as a product and every product as a series. As regards detail, this has to be done as follows: - 129. + u Ia + a,) is given, then this product, if we write n=1 n : 1a + a,) — Dns represents essentially the sequence (p,). This.sequence, on the other hand, is represented by the series tt O20 00h S(1+ay)...(1+a-1) a, n=2 This and the given product have the same meaning — if the product converges in accordance with our definition. But the series may also have a meaning without this being the case for the product (e. g. if the factor (1 }a,) is = 0 and all other factors are = 2). » @D 2. If conversely the series } a, is given, then it represents the n=1 . n . . . sequence for which 5, = Ya, This is also what is meant by the r=1 product 5 5 sy 7 a, ) Sie .-2 ...=0¢ - J] =a - J] 1 Es 1s 5 List St Linh + bly bl, ? — provided it has a meaning at all. And for this obviously all that we require is that each s,==0. In the case, however, of 5s, —0, although we call the series convergent with sum 0, we say that the product diverges to 0. I Thus e. g. the symbols : . = 1 ag. ifs a and = 1 » 2 et) 5 1 1 1 2 eED and 00+ FD): have precisely the same meaning. { i cain § 30. Connection between series and products. 299 It is, however, only in rare cases that a passage such as this from the one symbol to the other will be advantageous for actual investigations. The connection between series and products which is theoretically conclusive was, moreover, established by Theorem 8 alone, — or by Theorem 7, if we are concerned with the mere question of absolute convergence. In order to show the bearing of these theorems on general questions, we may prove — as analogue of Theorem S88, 1, and 89, 2, — the following: : © Theorem 11. An infinite product I1(1 + a) is unconditionally 130. convergent — 1. e. remains convergent, with value unaltered, however its factors be rearranged (v. 7, 3) — if, and only if, it converges absolutely?®. Proof. We suppose given a convergent infinite product I7(1 + a). The terms a,, certainly finite in number, for which |a, | > 1, we re place by 0. In so doing, we only make a “finite number of alterations” and we ensure | | < i for every n. The number m in the proof of theorem 8 may then be taken = 0. We first prove the theorem for the altered product. Now, with the present values of q_, Hla) and 2log(1+4a,) are convergent together, and their values U and L stand in the relation U == ¢L to one another. It follows that a rearrangement of the factors of the product leaves this convergent, with the same value U, if and only if the corresponding rearrangement of the terms of the series also leaves this convergent, with the same sum. But this, for a series, is the case if, and only if, it converges absolutely. By theorem 8 the same therefore holds for the product. Now if, before the rearrangement, we have made a finite number of alterations, and then after the rearrangement make them again in the opposite sense, this can have no influence on the present question, The theorem is therefore true for all products. Additional remark. Using the theorem of Riemann proved later (187) we can of course say, more precisely: If the product is not absolutely convergent and has no factor — 0, then we can by suitable rearrangement of its factors, always arrange that the sequence of its partial products has prescribed lower and upper limits » and Mu, provided they have the same sign as the value of the given product 1%, Here % and pu may also be 0 or + oo. ® Din, U.: Sui prodotti infiniti, Annali di Matem., (2) Vol. 2, pp.28—38. 1868. Y For a convergent infinite product has certainly only a finite number of negative factors; and their number is not altered by the rearrangement. 92928 Chapter VII, Intinite products. Exercises on Chapter VII. 85. Prove that the following products converge and have the values indicated: 2 pi-—1 2 ® 1\2n © Jor =v y ff (1+(5)") - 2 27-1 : ;) = 4 c Le —- 86. Determine the behaviour of the following products: ; ne 0 J+ 5) n=2 a log n IC ea (re Lo) i-1y) (14, Is al): 23 1 1 for «1, 5, go oe S6a. By 128, 2 the sequences 1 1 1 3 gpm lloyd. olor —logn and Yr=1fgrter “F—=logn have positive terms for n>1. Show that (x,|y,) is a nest of intervals. The value which it defines is again Euler's constant. 87. Show that II cos x, converges if 3 |x, |? converges. 88. The product in Ex. 86d has, for positive integral values of «, Go the value Vv 2. Hint: The partial product with last factor (1 re) is og — J 6-2] wii val, 89. Prove, with reference to Ex. 87, that cos Zeon conv $ 4 T 16 Ea (We recognise Viefa’s product mentioned in footnote * p. 218.) 90. Show, more generally, that for every x E08 E08 00% COR ns vs yay 2 4 8 16... =~ » h _ cosh cosh cosh T cosh = he = E eT le™% ef. g7% in which latter formula cosh =—rrp a sinhz=-—y3 ‘ 91. With the help of Ex. 90, show that the number defined by the nest of intervals in Ex. 8 ¢c Is = 2s where ¢ is defined as the acute angle for which cos =k. Similarly the number defined by Ex. 8d is 1 i = Bs x, , if ¥ is defined by coshd=-1, Exercises on Chapter VII. 229 29. In a similar way, show that the numbers defined in Ex. 8e and 8f have the values: sinh 2 ¢ $0 Yy e) —5g & with cosh? = Td sin 2 § ; 28. f) FN with cos?d = ie 93. We have 1-2 20mm) a pe ZO.) a, a,-a, a 0g... 0, =(1-%) (1-3)... (1-2). ay a, a, What can you deduce for the series and product of which we here have the initial portions? 94, With the help of theorem 10 of § 29, show that 1.3.5...22—-1) tg ol rr Vn 95. Similarly, show that, for 0 a, whose ferms a, are expressible 131. n=0 in the form a, =x, —=x, ,,, where x, is the term of a convergent sequence of known limit &, the sum of the series can be specified, for we have 2 >a, =1x,—§&. n=0 Proof. We may write : s,= @— 2) + @, — 2) (2, — By yg) = — 1, 4 Since x, — &, the statement follows. Examples. : 132. L. If « be any real number = 0, —1, —2,..., then (v. 68, 2h): $ 1 ¢ 1 1 nl =m) (c+n+1) 2. Similarly —=, as here a, = etn oT : 5 1 - 1 neole+n) (@+n+)(e+n+2) 2a(@+1) as here 1 1 “2 edn) (@t+ntl) (@Fntl)(etnt2d” 3. Senepally, if p denotes any positive integer, IL 1 > (+n) (e+n+1)...(a+n+p) 2 e(@+1)...(e+p—-1" 4. Putting « =}, we thus obtain, for instance, from 2.: 1 1 1 1 tomtreet Tw 5. Putting ¢ =1 in 3. we obtain 1 “ I. i] 2...6+1 '2:3...(p+2 Li or 2 1 pt ’ Borin)” p 241 /- [rd 934 Chapter VIII. Closed and numerical expressions for the sums of series. The following is a somewhat more general theorem. 133. O Theorem. If the term a, of a given series 2a, is expressible in the form x, — x, , where x, is the term of a convergent sequence of known limit E and gq denotes a fixed integer > 0, then Za, =X, + ofa dh F —gé. Proof. We have, for » > yg, S$, = (@ — 2) + (&, — Lad tr vob ll, =e) v8, =5,) i ral lo, Te dg = (2, +z, rhe mn (x ntl HHL es 1 Tp) Since x, — & (by 41, 9), the statement at once follows. Examples. 5 1 171 1 1 r Fo (ge ) 2 (ec +n) (ec +n-+9) q etait eo] ’ since here we have 1 ( 1 1 ) Cp=— —_— |. gen. atid In particular, wring c=1, 1 1 TIIRTRTII” [1 3 Teed gl) 9. For ¢ =1 and ¢ =2 we have accordingly: 1 1 1 3 vita and for «a =4, ¢=3: 1 1 1 23 nara Tn 3. Somewhat more generally, if %, as well as g, denotes a fixed integer >0: 1 : 0 tm) (@tntq..@@tntkyq : ig 1 5.E pret rtd letrti=19 4, Thus for a=1, g=2, k=2 we find stent samt Tm The artifices here employed may be extended to obtain, finally, - ] the following considerably further reaching : 134. OTheorem. If the terms of a series 2a, are expressible, for | every n, in the form B=, By 0,000 6 Wy, (R eONSIon, = 2) 3 where (x,) denotes a convergent sequence of known limit &, and the coefficients c; satisfy the condition 3 e,+ec+-+e6=0 Ls n § 32. Evaluation of the sum of a series by means of a closed expression. 235 then 2a, is convergent and has for sum: n= 08 le Tait at re bly fof one, J n= + (cg+2¢ +--+ k—1c)é. The proof is at once obtained by writing the expressions for @, %,; --,4,, one below the other so that terms involving a occupy the same vertical. Carrying out the addition in columns, — which of course is allowed even without reference to the main rearrangement theorem — we find, for m > k, taking into account the condition ful- filled by the coefficients c,, m r—1 k—1 ad; =o hg, tees Lom, +2 Ces ee de 0 Yr 00 which is again the sum of a finite number of terms. Letting m — oo, we at once obtain the required relation. Examples. ; n? ; 1. Putting Sub reo £=2, e;==1, ¢,=--1. we obtain 3 % 7 2n +1 2 REN BY tone ean 3 1 1 5 ( 1 as 1 Yell 2 oerner 271, = \—3+n+1 —j3+n+4l 84° These examples may of course easily be multiplied to any extent desired. 2. Application to the elementary functions. The above few theo- rems have, speaking generally, made us familiar with all types of series which may, without requiring any more refined artifices, be summed in the form of a closed expression. By far the most frequent series, in all applications, are those ob- tained by substituting particular values for x in series expansions of elementary functions and in series derived from these by every species of transformation or combination, or other known processes of deduction. Examples, obtained in this manner, of summation by closed expres- sions are innumerable, We must content ourselves with referring the reader to the particularly ample selection of examples at the end of this chapter, in the working out of which the student will rapidly be- come familiar with all the main artifices used in this connection. The developments in this and the following section will afford further guidance in this part of the subject. Let us merely observe quite generally, for the moment, that it is often possible to deal with a given series by splitting it up into two or more parts, each of which again represents a convergent series; or else by adding to or subtracting from IX, ,» term by term, a second series of known sum. In particular, if 4, is a rational function of n its expansion in partial fractions will frequently be a considerable help. 135. 236 Chapter VIII. Closed and numerical expressions for the sums of series. 3. Application of Abel's theorem of limits. A further means of evaluating the sum of a series, — one of great theoretical importance, differing from that just indicated in the principle it involves, though in most cases intimately connected with it in virtue of 101, — con- sists in applying Abel's theorem of limits. Given a convergent se- ries 2a,, the power series f(x)= 24,2" converges at least for —1<2< +1, and hence, by 101, =a = Ym flo). If we suppose that the function f(x) which the power series represents is so far known, that the latter limit can be evaluated, the summation of the series is achieved. The developments of Chapter VI offer a wide basis for this mode of procedure, and in fact Abel's theorem has already been used there more than once in the sense now explained. We shall give here only a few relatively obvious examples, with a reference to the exercises at the end of this chapter. a We are already Ep with the series: Ldn rez 2 & J = at : = T = lim 1 n = lim tang =—. a 0 2n+1 z>1— ea 2041 451-0 - 4 +1 = Bid log (142) =1og 2, >1-0 We have the further example 0 — nH» : xt x7 : a EL a Zs) Asn [Be fey] The series inside the bracket has for derived series 1—af4af—fee=— 1 T+ a8 and therefore represents the function (v. § 19, Def. 12) Zz 5 dz +1 (x + 1)2 2 -Liw «122ml, 7 0 +2 "8 ea ti’ 3 = oan Accordingly, the sum ot the given series is = 1 5 log 24+ —— 3 5 4. Similarly we find (v. 3 » Def. 12) =H 1 i eel] —g5t- pas SVE [+10g G +22). For further series constructed on the same the formulae of course become more and more complicated. 4. Application of the main rearrangement theorem. Equally great theoretical and practical significance attaches, in our present problem, to the application of the main rearrangement theorem. This application we proceed at once to illustrate by one of the most important cases; additional examples will again be furnished by the exercises. In 115 and 117, we obtained two entirely distinct expansions of the function z cot, both valid at least for every sufficiently small |z|. § 32. Evaluation of the sum of a series by means of a closed expression. 237 If, in the first of these, we replace x by mx, we obtain, certainly for every sufficiently small 1, it Sr Wy fem) — I= Each term of the series on the right may obviously be expanded in powers of x: 2 a? -— $2 (ZY. (B==1,2,,.. fized) SS og% a8 2 k x — k These are the series 2 of the main rearrangement theorem; since the series {® of that theorem in our case only differ in sign from the series z® themselves, the conditions of that theorem are all fulfilled, and we may sum in columns. The coefficient of £2? on the right then becomes -2 2 . (p fixed) and since, by 97, it has to coincide with that on the left, we obtain the important result (once more denoting the index of summation by n) 51 2p 2 0)°P i = 5 hg = (—1)?~ Te . (p fixed) 136. This gives us the sum of the series 1 ; 1+ toto te (p fixed) in the form of a closed expression, since the number z and the (ra- tional) Bernoulli's numbers may be regarded as known. In perio, 1 x? & Zi b7 y — DS === — ==. 4 rent 6 nai pt 802 = nt 4 4 Quite incidentally, formula 136 shows that Bernoulli’s numbers B,, are of alternating signs and that (— 1)»—1B,, is positive; furthers that they increase with extreme rapidity as # increases; for since the value of x lies between 1 k= and 2, whatever be the value of #, we necessarily have 2 (2m)! rad 2(2n)! i aa Ben> gta? whence it follows that Bats — +00. Finally, as the above transformation 2n holds for |x| <1, it also follows that the series 115 converges absolutely at least for |x| < mz. But for |2|> = it certainly cannot converge absolutely, for then cotz would be continuous for z =z, by 98, 2, which we know is not the case; thus the series 118 has exactly the radius m. It follows from this that 116 a has the radius Z , 116 b the radius =. 137. 138. 9238 Chapter VIII, Closed and numerical expressions for the sums ot series. It is not superfluous to try to realise all that was needed to ob- tain even the first of these elegant formulae3. This will be seen to involve a very considerable portion of the whole of our investigations up to this point. The above provides us with the sum of every harmonic series with an even integral exponent; we know nothing yet of the sum of an harmonic series with odd exponent (> 1); that is to say, we have : . 1 ; not succeeded as yet in connecting such a sum ( g. poy with any numbers occurring elsewhere. (There is of course no obstacle to our evaluating the sum of any harmonic series numerically, to any degree of approximation®; v. § 35). On the other hand, our results readily yield the following Be formulae: We have eel LE +2 a The latter series is precisely the same thing as % > I Subtract- Tp a 8 ing this from both sides, we obtain Z 1 1 zx = = —=1— = 2rn—-1)%? ( oF 77) 3 > 1 n3P or vor =(—1)P—1 1+ += nr 9D So Bay For p=1,2,3,..., the sums are in particular gt rmb ad 82 96°. 960° If we again subtract the same series ih ES , we obtain 922 n=1 n=? ger 2y 0 oS alt 2 n2P ro n?P or 1 1 1 2%r—1_y — mi wits nie Sakis es V0 i fe =1 a2 5 James and John Bernoulli did their utmost to sum the series Tid ni 1 +3+tgtygt The former of the two did not live to see the seclution of the problem, which was found by Euler in 1736. John Bernoulli, to whom it became known soon after, wrote in this connection (Werke, Vol. 4, p. 22): Atque ita satisfactum est ardenti desiderio Fratris mei, qui agnoscens summae huius pervestigationem difficiliorem quam quis putaverit, ingenue fassus est omnem suam industriam fuisse elusam ... Utinam Frater superstes esset! A second proof, of a quite different kind, will be found in 156, a third in 188, and a fourth in 2X0. @® 8 T. J. Stieltjes (Tables des valeurs des sommes S, = > n=1 matica, Vol. 10, p. 299, 1887) evaluated the sums of these series, up to the ex- ponent 70, to 32 places of decimals. 1 —» Acta mathe- n % § 32. Evaluation of the sum of a series by means ot a closed expression. 239 In pariicolar, for p==1,2, 3,... the isums’ are 1 0 0 a 31 6 B27 mmr Here again, however, we know nothing of the corresponding series with odd exponents. — The last two results might of course also have been obtained by starting with the expansions in partial fractions of the functions tan or = and reasoning as above for that of the func- tion cot. We may deduce further results by treating the expansion in partial fractions, given in 118, of the function or Ie F 7 pant 3 5 2-1 2r+D ICL LET rr a = 2 Cv+1)2—z®" 2 The »®™ term is here expressible by the power series (~ after rearranging, the coefficient of 22? thus becomes: Soi GO 0, 2 ay Sm te aie 2 © z k I Crit? Let us denote these sums provisionally by Oy p+15 then EE —¢, 0,2 0 0 + eee 4cos”” 2 or 1 4 24\2 22\4 : cosz Hote, = thos &) to] : On the other hand, this power series may be obtained by direct division and its coefficients — just like Bernoulli's numbers in 103, 5 — by simple recurring formulae. We usually write 335 so that (-f thE Rr + Be — + =, This gives E,=1, and, for every # >1, recurring formulae’ which may be written as follows (after multiplication by (2 n)!): Brut (C1 Bons + PVT 0, 139. 7 The numbers determined by these formulae, (which are moreover rational integral numbers) are usually referred to as Euler's numbers. The first 30 have been calculated by W. Scherk, Mathem. Abh., Berlin 1825. 140. 24(0 Chapter VIII. Closed and numerical expressions for the sums of series. or in the shorter symbolical form (cf. 106): (E+ 1)" + (E—1y-=0, now holding for every £ >1. We deduce without difficulty: E,—E,—E,—---=0 and E,=1, E,=—1, E,=5, FE, = —61, E,==1385,..., In terms of these numbers, which we are perfectly justified in con- sidering as known, we have, finally, (— 1)? Pe? its (zp)! a pic ep’ ie. I» 1 1 lt = Dl io mat 527+1 =} ( ) 2272+2 (2 p)1 In particular, for =0,1,2,3,..., this gives the values 1 1s Bs 61 : 1%: m%y. RE7 ml for the sums of the corresponding series. § 33. Transformation of series. In the preceding section (§ 32), we became acquainted with the most important types of series which can be summed by means of a closed expression — either in the stricter or in the wider sense of the term. In the evaluations last made, which are really of a profound nature, the main rearrangement theorem played an essential part; indeed, in virtue of this theorem, the original series was changed, so to speak, into a completely different series which then yielded further informa- tion. We were therefore principally concerned with a special frans- formation of series 8. Such transformations are frequently of the greatest use, and indeed even more so in the numerical calculations which form the subject of the following two sections, than in the determina- tion of closed expressions for the sums of series. To these trans- formations we will now turn our attention, and we start at once with a more general conception of the transformation deduced from the main rearrangement theorem and repeatedly applied to advantage already in the preceding section. 8 Such transformations were first indicated by J. Stirling (Methodus diffe- rentialis, London 1730); they are based, in his case, on similar lines to the above, excepting that he fails to verify the fulfilment of the conditions under which the processes are valid. > = § 33 Transformation of series. 241 0 Given a convergent series X:®, let each of its terms be expressed, in any manner, (e.g. by § 32, p. 232) as the sum of an infinite series: 209) en 2,2 a 2, 1 2 + i . +4 2. ® be a | : Biph 20 + a te 2,2 + Si + 2 2 + ae |= a +a, 4 a,P fess +a B eines We shall assume further that the vertical columns in this array them- selves constitute convergent series, and denote their sums by sO, sD, 8%, .... Under what conditions may the series > £9 n= formed by these numbers be expected to converge, with 2 ey = $M; ? n=0 If this equality is Ta we have certainly effected a trans- formation of the given series. The main rearrangement theorem im- mediately gives the © Theorem. If the horizontal rows of the array (A) all constitute absolutely convergent series and — denoting by {® the sum, 3 a, ® |, of the absolute values of the terms in one row —, if the series Zc® gs convergent, the series 2 s™ also converges and = 2% 2®, It is this theorem that we have applied in the preceding para- graph. The question arises whether its requirements are not un- necessarily stringent, whether the transformation is not allowed under very much wider conditions. A. In this direction, an extremely far-reaching theorem was proved by A. Markoff®. He assumes first only that the series constituted by the vertical columns of the array (A) converge, as well as the original series and the series cons'ituted by the horizontal rows of the array. The @® 0 numbers s® have thus determinate values. Since Y 2% and 3 q,® k=0 k=0 0 converge, so does DEW — a,™); and also, similarly, for any fixed mm, E=0 the series > OP oP = a oz ver gf) (m fixed). =o ? Mémoire sur la transformation de séries (Mém. de I’Acad. Imp. de St. Pétersburg, (7) Vol. 37. 1891). Cf. a note by the author, “Einige Bemer- kungen zur Kummerschen und Markoffschen Reihentransformation”, Sitzungs- berichte der Berl. Math. Ges., Vol. 19. pp. 4—17, 1919. 141. 142. ‘949 Chapter VIII. Closed and numerical expressions for the sums of series. The terms of this series are, however, precisely the remainders, each with the initial index m1, of the series constituted by the individual rows of the array. If, for brevity, we denote these remainders by a so that P= > 2 (2 and m fixed), n=m “a the series MB R, (m fixed) £=9 is convergent. The further assumption is then made that R,—0 when wm — 00. It may be shown that under these hypotheses 3s™ converges and = 3 :®. The theorem obtained will thus be as follows: O Markoff’s transformation of series. Let a convergent series 32% be given with each of its terms itself expressed as a convergent b=0 series: (A) = a,® 4 a,® Lo ot 22 Ave (B=0,1,2..)) Let the individual columns 3 a,® of the array (A) so formed represent k=0 convergent series with sum s%, n=0,1,2,..., so that the remainders yD 3 2 (m > 0) n=m of the series in the horizontal rows also constitute a convergent series +2 7 = R,, (m fixed); F=0 and suppose finally that, with increasing m, the numbers R-—0: In that case the sum s™ of the series in vertical columns also forms | a convergent series, and we have Ss —= 320, n=>0 E=0 The proof is extremely simple. We have () (®) *) bez0,1,2, 004 Go — tpn — "nia n=012... Accordingly 24a Wn sgl? = (rm +r eel) = ads Fm Fee Fab): 10 Here we of course take m = 0 to give the whole series, i.e. 2B itself. © § 33. Transformation of series. 243 On the right hand side, we now have the initial portions of two series, convergent by hypothesis. Letting ¢ increase beyond all bounds, we deduce at once that, for fixed #: TaD msm B= By But this implies (a) sO+sWt.eq4s®=R —R_,,; it follows, by our second hypothesis, that 3 s™ converges, with ei, es =R n=0 ; WL Fm Now R, is precisely = 7,t = z2*”; thus we have proved all that was required. Besides this, our equality (a) shows that it is necessary, as well as sufficient, if 2 s™ is to converge, that R, should tend to a finite limit, and that this limit should be 0, if X's™ is to equal 3z®, B. The superiority of Markoff's transformation over Theorem 141 consists, of course, in the absence of any mention of absolute conver- gence, only convergence pure and simple being required throughout. Its applications are numerous and fruitful: those bearing on numerical evaluations will be considered in § 35, and we shall only indicate in this place one of the prettiest of its applications, which consists in obtaining a transformation given by Euler! — of course, in his case, without any considerations of convergence. It is advantageous here to use the notation of the calculus of finite differences, and this we will accordingly first elucidate in brief. Given any sequence (m,, ®,, #,,...), the numbers Boi Tyr, Tyg = Bpy veri By = Wp gy ave are called the first differences of (x,) and are denoted by dass Ado, ..., dz, ... The differences of the first order of (4z,), i. e. the numbers dx, — d=, , ,, k=0,1,2,..., are called the second differences of (x), denoted by AZ, mys ainsi 0; eee In general, we write for n > 1, dP ig = An =A Ae ==0,1,2,..,) and this formula may also be taken to comprise the case n= 0 if we interpret 4%x, as being the number z, itself. It is convenient to imagine the numbers «, and 4”, arranged in rows so as to form the following triangular array, in which each difference occupies the 11 Institutiones calculi differentialis, 1755, p. 281. 143. 144. 944 Chapter VIII. Closed and numerical expressions for the sums of series. place in its own row immediately below the space, in the row above, between the two terms whose difference it is: a Z Ta: 0° 1° 3 Ady, du dry, Any... . (4) Mozy, Ba, Az, .. 5% : die., Aion, ..... ; B25 The difference 4" xz, may be expressed in terms of the given num. bers x, directly. In fact 2 — = he — Be =dr, — Ax, =@ — 3) — (1; — Tyo) = — i 1 = = 28 Tray and similarly 3p = a : HBr, =z —35,, 13x ,—2,,;3 the formula n n n fT 4 wpm e— (7) npr + (5) @rrr—+- + (=D (3) itn for fixed %, is thus established in the cases # = 1, 2, 3. By induction, its validity for every n follows. For, supposing 143 proved for a par- ticular positive integer n, we have for »n -}- 1: dvi = A"g —A"n ,, =r (3) Try 1 & Teper th erohl=1) 4 )e tw EO ee, += pre” Vonnis whence by addition, since (3 +( ® )= £5 we have the for- mula 148 for # 1 instead of #. This proves all that is required. y—1 Making use of the above simple facts and notation, we may now state the following theorem: O Euler's transformation of series. Given an arbitrary con- vergent series — dnd 19 Fe 1) ©“ =% a, = ay — - ’ we invariably Ph @®» @® A n a Se ce ) k=0 n=>0 ant 1 12 The series need not be an alternating series, i. e. the numbers a, need not all be positive. There are however small, though by no means essential, advantages in writing the series in alternating form as above, when effecting the transformation. § 33. Transformation of series. 245 i. e. the series on the right also converges and has the same sum as the given series. Proof. In the array (A) of p.241, we substitute for =; k k[1 1 (b) 02 = (1) 4" 0, — gar Ama]. By 131, if we now sum for every #, keeping % fixed (i. e. form the sum of the k&™ horizontal row), we obtain = 3 0® = (~ Ppa =(= De, (fixed) For n n ; War (8) = (1) ans t = +0" (3) tug lim 5 lim gw 2 n—>wo n—>w is equal to zero by 44, 8, because q4,, — Buys Hpgs +» OCTialy form a null sequence. Accordingly (b) gives an expression for the in- dividual terms of the given series 3 (— 1)" a, in infinite series. Forming the sum of the n't column, we obtain the series © 1 1 i= 9:1 A20, = os Arle, | (n fixed); . the generic term of this series, as 4? +1q, = A4"a, — 4" a, ,, can be written in the form 1 =. ,, n n irda, + 4" a] = os [(— DF A" a, — (— DFA, ], so that the series under consideration may again be summed directly, by 131. We obtain J i he [4" ay — lim (— 1)* 4" a,], (n fixed). k=0 k>w» Since, however, the numbers 4, form a null sequence, so do the first differences and the nth differences generally, for any fixed n. The vertical columns are thus seen to constitute convergent series of sums s® 42a, Tout The validity of Euler's transformation will accordingly be established when we have shown that R, — 0. Now the horizontal remainders are seen to have the values A 1 47 ay, YY selfs ) gm ’ 13 The proof that the transformation was always valid, provided only the series 3 (—1)* a; is assumed also convergent, was first given by L. D. Ames (Annals of Math., (2) Vol. 3, p. 185. 1901). Cf. also E. Jacobsthal (Mathem. Zeitschr,, Vol. 6, p. 100. 1920) and the note bearing on that by the author (ibid. p. 118). 9% 246 Chapter VIII. Closed and numerical expressions for the sums of series. — following precisely the same line of argument as was used above for the entire horizontal rows. Thus 1 0 = M(—1)*4mq, (fixed mm). 2™ =o If we write for brevity \k (Dt tgs Ter, this series for R, may be thought of as obtained by term-by-term = addition from the (m +1) series: m m m rs Us nie ot (fr (8) (2) m 9m 2 Hence therefore, as » is the term of a null sequence, so is R, , by 44, 8. This proves the validity of Euler's transformation with full generality. Examples. 1. Take To bed i=l-at 2 The triangular array (4) takes the form 1 1 1 1 1 ’ 9 ) 3? a’ iT eo 1 1 1 1 ¥3 98>. 39 4% 1.2 1.2 12 123 234 345 °° . ee Tew fy The general expression of the n'! difference is found to be aa nl! te CT DC+D-- Cat) so that in particular 1K n Be At ay = il This is easily verified by induction. Accordingly we have 1 1 1 1 1 va 1 feb gly—yt— “Inti mt The significance of this transformation e. g. for purposes of numerical calcu- lation (§ 34) is at once apparent. 2. With equal facility, we may deduce 4 1.1 1-2.3 T=t—gt+3—7t+—=z[1+3 ti i 5 sete] Tn what cases this transformation is particularly advantageous for pur poses of numerical calculation will be seen in the following section. § 34. Numerical evaluations. 247 C. OKummer’s transformation of series. Another very obvious transformation consists simply in subtracting from a given series one whose sum is capable of representation by means of a known closed expression and which at the same time has terms as similar in con- struction as possible to those of the given series. By this means, 2 L. subtracting for instance from s = Zs the known series (v. 68, 2b) bas iw w=1" we deduce the amr sm 2 mth Sito The advantage of this transformation for numerical purposes is at once clear. Simple and obvious as this transformation is, it yet forms what is really the kernel of Kummer's transformation of series'*; the only difference being that a particular emphasis is now laid on a suitable choice of the series to be subtracted. This choice is regulated as follows: Let 2a, —s be the given series (of course, by hypothesis, convergent). Let X¢ = C be a convergent series of known sum C. Let us suppose that the terms of the two series ave asymptotically proportional, say lim = = y + 0. n—>wo 7 In that case an > c, =) a,=vC+ 5 (1—r 2) an, 145. =0 and the new series occuring on the right may be regarded as a trans- formation of the given series. The advantage of this transformation lies mainly in the fact that the new series has terms less in absolute value than those of the given series, as in fact (1 y 2) —0. Con sequently its field of application belongs for the most part to the do- main of numerical calculations and examples illustrating it will be found in the following paragraph. § 34. Numerical evaluations. 1. General considerations. As repeatedly explained already, it is only on very rare occasions that a closed expression, properly so- called, exists for the sum of a series. In the general case, the real 14 Kummer, E. E.: Journ. f. d. reine u. angew. Math., Vol. 16, p. 206, 1837. Cf. also Leclert and Catalan (Mémoires couronnés et de savants étrangers de I'Ac. Belgique, Vol. 83, 1865—67) and the note by the author mentioned in foot- note 9. 9248 Chapter VIII. Closed and numerical expressions for the sums of series. number to which a given convergent series, or the sequence of num- bers for which it stands, converges, is, so to speak, first defined (given, determined, ...) by the series itself, in the only sense in which a number can be given, according to the discussion of Chapters I and II1%. In this sense, we may boldly affirm that the convergent series is the number to which its partial sums converge. But for most practical purposes we gain very little by this assertion. In practice, we usually require to know something more precise about the magnitude of the number and to compare different numbers among themselves, etc. For this purpose, we require to be able to reduce all numbers, defined by any kind of limiting process, fo one and the same typical form. The form of a decimal fraction is that most familiar to us to-day, and the expression, in this form, of numbers represented by series accor- ~ dingly interests us first and foremost!®. The student should, however, get it quite clear in his own mind that by obtaining such an expres- sion we have merely, at bottom, substituted for the definition of a number by a given limiting process, a representation by means of another limiting process. The advantages of the latter, namely of the decimal form, are mainly that numbers so represented are easily com- pared with one another and that the error involved in terminating an infinite decimal at any given place is easily evaluated. Opposed to this there are, however, considerable disadvantages: the complete ob- scurity of the mode of succession of the digits in by far the greater number of cases and the consequent labour involved in their succes- - sive evaluation. These advantages and disadvantages may be conveniently illustrated by the two following examples: TT (F=)1—g+g—7+—-=0185395... 1 ll 1 (log2 =)1 — 5 +5 — 4 + — +» = 0693147 .... By the series, distinct laws of formation are given; but they afford us no means of recognizing which of the two numbers is the larger of the two, for instance, or what is its excess over the smaller number. The decimal fractions, on the other hand, exhibit no such laws, but give us a direct sense of the relative and absolute magnitudes of both numbers. 15 Indeed an infinite series — our previous considerations. give ample confirmation of the fact — is one of the most useful modes of so defining a number, one of the most significant both for theoretical and practical purposes. 18 And only in special cases the expression in ordinary fractional form. The reason is always that of convenience of comparison; which, of 1% or 33, is the larger, we cannot say at once, whereas the answer to the same question for 0 647 and 0-641 requires no calculation whatever. . § 34. Numerical evaluations, 249° ~ We shall therefore henceforth reserve the term numerical eva- luation for the expression of a number in decimal form. As no infinite decimal fraction can be specified in foto, it will be necessary to break it off after a definite number of digits. We have still a few words to say as to the significance of this process of breaking off decimal fractions. If it be desired, for instance, to indicate the number ¢ by a two-digit decimal fraction, we may with equal justification write 2:71 and 2-72, — the former, because the two first decimals are actually 7 and 1, — the latter, because it appears to involve a lesser error. We shall therefore make the following con- vention: when the 7 specified digits after the decimal point are the actual first n digits of the complete infinite decimal which expresses a given number, we shall insert a few dots after the nt? digit, writing for instance ¢ = 2-71...; when, however, the number is indicated by the nearest possible decimal fraction of # digits, we insert no dots after the n * digit, but write e. g. e /&~ 2:72 1%; in the latter case the nth digit written down is thus the nth digit of the actual infinite fraction raised or not by unity according as the succeeding part of the infinite fraction re- presents more or less than one half of a unit in the #®™ decimal place, In point of fact, cither specification has the effect of assigning an interval of length 1/10" containing the required number. In the one case, the left hand endpoint is indicated, in the other, the centre of the interval. The margin, for the actual value, is the same in both cases. On the other hand, the error attaching to the indicated value, relatively to the true value of the number considered, is in the former case only known to be 1/10", in the latter < 1/10". For this reason, we may describe the first indication as theoretically the clearer, and the second as practically the more useful, of the two. The diffi- culty of actual determination of the digits is also in all essential par- ticulars the same in both cases. For in either, it may become ne- cessary, when a specially unfavourable case is considered, to diminish the error of calculation to very appreciably less than 1/10" before the nt digit can be properly determined. If we are, for instance, concerned with a number «= 527999999326..., — to determine whether ¢ = 5-27... or 528... (retaining twp decimals), we have to diminish the error to less than a unit in the A? decimal place. On the other hand, if we are concerned with a number f= 2:3850000026..., the choice between f~~2:38 and 2:39 would be influenced by an uncertainty of one unit in the 8th decimal place!8. 7 In e=271..., the sign of equality may be justified as representing a limiting relation. ~ 18 The probability of such cases occurring is of course extremely small, ~ By mentioning them, we have merely wished to draw attention to the signi- « ficance of these facts. In Ex. 131, however, a particularly crude case is indicated. 950 Chapter VIII. Closed and numerical expressions for the sums of series. 2. Evaluation of errors and remainders. When given a conver gent series Yq =s, we shall of course assume that the individual terms of the series are “known”, i. e. that their expressions in decimal form can easily be obtained to any number of digits. By addition, every partial sum s, may accordingly also be evaluated. The question remains: what is the magnitude of the error attaching to a given s 71? Here the word error designates the (positive or negative) number which has to be added to s, to obtain the required value s. Since this error is s —s,, i. e. is equal to the remainder of the series, starting im- mediately after the nth term, we will denote it by #,, and the process of determining this error will also be designated by the term evaluation of remainders. In practical problems, evaluations of remainders almost invariably reduce to one of the two following types: A. Remainders of absolutely convergent series. lf s = 2a, con- verges absolutely, determine a series Xa,’ of positive terms, capable of summation in a convenient closed expression, and with terms nof less than the absolute values of the corresponding terms of the given series (though also exceeding these by as little as fo Obviously | 7. [Sia ltl. + r ZT Aer 7 C ays =7,/ and the number 7’, which is assumed known, thus fiviEin a means of estimating the magnitude of the remainder 7,, i. e. 7, |<), and § this all the more closely the less a, exceeds [os jk: A particularly fremont case is that in which, for some fixed m, and every k > 1: ; Ap + k |< | @, a, in that case, of course, 2 a im | = | a, 1-2 and in particular, if 0 10%, but an evaluation of a million terms is, for practical purposes, quite impossible. The rapidity of the conver- gence may be increased very materially by Euler's transformation 144, 2. In the next paragraph, we shall discuss the utility of such transformations for purposes of numerical calculation. Our present object is to deduce more convenient series expressions for z directly from the tan™?1 series itself. The series expansion for tan~?! + = > is already of appreciable ’ - VY use: this gives 2 Cf. p. 249. § 34. Numerical evaluations. 238 The following mode of procedure, however, provides considerably more convenient series? The number 1 17 3 1 1 eR ee so, a th == 150" dp om ee TE ETRE = Er AE is easily calculated from the series itself (see below). For this value of ¢, tang =1, and so 2 tan o 5. ltr and 120 tan 4 = 119 . Consequently 4 ¢ exceeds = by only a small amount. Writing J 4 — a mm Bs we have tan4 ¢ —tani x 1 tan f = I+tandotaniag 239° Hence f can very easily be evaluated from the series 1 1 Lo 0)" 00h mm =e The two numbers « and fj give us n=4(4a—p) 1 1 1 1 1 gto] df mofo ot fol fren] 146. > 55° 239 3.2395 B= tan] If it be desired to obtain the first seven true decimals of x, we may endeavour to attain this end by taking, say, 9 decimals for each of the terms and for the remainder? — a scanty enough margin, for the errors incurred on the numbers ¢ and # have ultimately to be multiplied by 16 and 4 respec- tively. Denoting the first series by a, —a,+a;—-..., the second by a’ —ay+a'—+-.., and the corresponding partial sums by s, and 5, the calculation proceeds as follows: a, =0200000000 - a, = 0002666667 a; =0-000064000 a, =0000001829— a, =0r...... 57 B= 0 0 0 a, +a; + ay = 0200064057 ata, +a, =0002668498——— S53 =0197895559% ++ 28, 0 aiD eT Tare Ey TT aE we are thus able to assert that our sum s satisfies the conditions 1 1 1 1 1 xt Il tmt tar, pets itm the, for every n. To secure 6 decimals, we may accordingly need only 1000 terms. This is still too large a number for practical purposes. But in special examples this method of upper and lower estimates of the remainder (cf. Ex. 131) may lead to a satisfactory result. These cases are, however, so rare, that they do not come into account for practical purposes. Greater importance attaches to methods for transformation of slowly convergent into rapidly convergent series, because they admit of a far wider range of applications. To these methods we proceed to give our attention. § 35. Applications of the transformation of series to numerical evaluations. In cases of slow convergence, one naturally attempts to change the given series into one with a more rapid convergence, by means of some suitable modification. We proceed to examine in this light the transformations discussed in § 33, so as to see how far they will be of use to us here. A. Kummer’s transformation. For this transformation it is im- mediately obvious whether and to what extent an increase in the ra- pidity of the convergence can be obtained by it. In fact, using the notation of 149, we have @® 0 Cn 2, =30C 4. (1—72)a,; n=0 n=0 2 as (1 — y 22) —0, the terms of the new series (from some index on- wards) are less than those of the given series. The ‘method will ac '§ 85. Applications of the transformation of series to numerical evaluations. 261 cordingly be all the more effective the smaller the factors (1— r=) are, from the first; or in other words, the nearer the terms of Jc, are to those of 2a. Examples. 153. 1. We found on p. 247 that 3 =1+ Dwr The terms of the new series are asymptotically equal to those of the series , 1 leg 1. = 1. A 2 rrr SED TOE i? thus here TA and y =1, and so 1 ol 2 ee + 3 ry Proceeding in this manner, we obtain, at the pth stage: @ iq 1 2 -1rbrht rien Seren n=1 ~ The latter series, even for moderate values of p, shows an appreciably rapid convergence. 2. Consider the somewhat more general series oo So 1 Ti cavbitrary £ 0, 1 ... 2 2 GTC Te TD MTT P=) |p, integer 21 Here we take = (149) ay— (1 +149) ans1, 2=0,12..., and we try to determine y (independent of #) so that ¢, is as near a, as possible?. Here we have C=y-a, and a simple calculation gives y 5 Hence we obtain yn fs ttn hl Stair Er -T=, ta. The expression in the large bracket is 1_PENO+atp)—@t1+) (ta)? @p—1)(n+etp) Since, by simplification, the terms in #3 and #® must disappear of themselves, this gives Qa+8p—2-2y)pn+2p—1—y) ade ln @p—1(m+a+p)? If we now choose y so that the terms in z also disappear, i. e. take y=atop -1» then the expression in the large bracket above now becomes 2 1 2@p—-1) (n+a+p)?’ % The choice of a number c, of the form x, — zp +1 will, by 131, always prove most convenient, as in that case C at any rate may be specified at ‘once and the choice still be so arranged that the ¢,’s are near to the a,’s. 154. 262 Chapter VIII. Closed and numerical expressions tor the sums of series. and accordingly 5 1 3 1 (et 5-1) o ‘ = Er) @tr=D 2@P=D 2 WFD Tet hE The transformation thus has the effect of introducing an additional quadratic factor in the denominator. — Particular cases: 2)ae=1. se BIEL. FP 3 1 3m BY iw 1 EE... TI0 0,2 Gr GIGI TIe Write for brevity 3 ! = aaa pe REG... Lp -1 TT =n (LD) TE the result then takes the form 5 Bf hE 2 SCs -D rr og, TD TY This formula enables us easily to obtain very rapidly convergent series for 1 Ss = Sh = Pr . b) Similarly, for « = 3 1 = Crt ECa+3)...Cnr2p—1° 3p—1 1 2p8 = 1 S2Rp-D TB... @p=D2 2p—1,2 Eni Dt...@nr pF This formula similarly leads to rapidly convergent series for 2 aoe For further examples, see Exercises 127 seq. . : B. Euler’s transformation. Euler's transformation 144 need not by any means involve an increase in the rapidity of convergence®® of the series to which it is applied. ® n : For instance the transformation of J (3) gives the series n=0 3 oe n 3 ] 3» (3) , which evidently has a less rapid convergence. But even 3% The explicit definition of what we mean by more or less rapid con- vergence will be given in § 87: Xa, is said to be more or less rapidly con- vergent than Xa,, according as ’ ’/ ’ m|_ item uBR ob AE Tn uti +g: § 85. Applications of the transformation of series to numerical evaluations. 263 in the case of alternating series, the effect need not be an increased rapidity of convergence; indeed the following three examples show that all conceivable cases may actually occur here: 1 > Le a x 2 (—1) 5s gives a more rapidly convergent series, eo ihas 2, x (— Pw » 5» series with the same rapidity of conver- n=0 ee gence, Pa . : I = /5\2 3. = (—1)" = » is Jess rapidly convergent series, 7S 5) . We “hall now show, however, that such an increasc in the rapi- dity of the convergence does result, in the case of those alternating series 2 (—1)"a,, a, > 0, whose terms, though not showing rapidity of convergence, still tend to zero in a particular regular manner, which we proceed to describe. These are the only types of alternating series of any practical importance. The hypothesis required will be that not only the numbers a form a monotone decreasing sequence, i. e. have positive first diffe- rences Aa,, but that the same is true of all differences of every order. A (positive) sequence a, a,, a,, -.. is said to have p-fold monotony?’ if its first, second, ..., pt differences are all positive, and it is said to be fully monotone if all the differences A257 Gin=0,12, er) are positive. With these designations, the theorem referred to is: Theorem 1. If 3 (—1)"a, is an alternating series for which 153. n=0 the (positive) numbers ay, a,,... form a pel monotone null se- quence, while, from the first, Pry =a > (for every n)®S, then the : 1 transformed series end ay will converge more rapidly than the given series. The proof is very simple. As Hoty =a, we have 4. > a,-a Further, for the remainder 7, of the given series we have (Drs, =a mh mda, dad Aa en hence, since (A is itself a monotone null sequence, [7 2% (A aysy + dtp day y+) =a, Ls ; drat. 37 Cf. Memoir of E. Jacobsthal referred to in 144. 38 This assumption is the precise formulation of the expression used above, that the given series should not converge particularly rapidly. The series will in fact, as the example shows more distinctly, converge less rapidly than = (5) 9264 Chapter VIII. Closed and numerical expressions for the sums of series. On the other hand, as 4"a, — 47t1a,= A" a, > 0, the numerators of the transformed series alto form a monotone null Sequence, and in particular are all ’ wh were anticipated in 144. For further appli- cations it is essential to know which null sequences are fully monotone. We may prove, in this connection, by repeated application of the first mean value theorem of the differential calculus (§ 19, Theorem 8), the following theorem: Theorem 2. A (positive) sequence a, a,, ... is fully monotone decreasing if a function f(x) exists, defined for x => 0, and possessing differential coefficients of all ovders for x > 0, for which f(n)=a, while the kth derived function has the constant sign (—1)k (k=0,1,2,...) Accordingly the numbers a’ (0a); tcons &>0,0> 0); I ee P=... (n+ p)* log (n+p) for instance, form fully monotone decreasing sequences; and from these many further sequences of this kind may be deduced, by means of the Theorem 8. If the numbers ay, a, ... and by, b,, ... constitute fully mono- tone decreasing sequences, the same is true of the products ay by, a,b, a,b,, : Proof. The following formula holds, and is easily verified by induction relatively to the index k: k A%a,b, = x (Brean, r=0 g It shows that, as required, all the differences of (a, b,) are positive, if those of (a,) and (b,) are so. The following may be sketched as a particular numerical example: The series © rr ad 1 1 200 ERED Emme § 85. Applications of the transformation of series to numerical evaluations. 265 has extraordinarily slow convergence; in fact, it converges with practically as small a rapidity as Abel's series 2 1[n (logn)?. Yet by means of Euler's transformation, its sum may be calculated with relative ease. If we use only the first seven terms (to inclusive) we can deduce the first seven terms of the trans- 1 log 16 formed series. If we use logarithms to seven places of decimals, we find, with 6 decimals secured, the value 0:221840... for the sum of the series? C. Markoff’s transformation. As the choice of the array (A), p.241, from which Markoff's transformation was deduced, is largely arbitrary, it is not surprising that we should be unable to formulate general theorems as to the effect of the transformation on the rapidity of the convergence. We shall therefore have to be content with laying down somewhat wider directing lines for its effective use, and with illustrating this by a few examples: Denoting as before by Xz® the given series (assumed convergent), we choose the terms of the O column in our array (A) to be as near as possible to those of the given series, and at the same time to possess a sum s© which we can indicate by a convenient closed expression; this is analogous to the condition of Kummer's trans- formation. The series X(z® — ¢,*) now certainly converges more rapidly than 3 z®; proceed with this new series in the same way, for the choice of the next column in our array, and so on. The effect of the transformation will be similar to that of an indefinitely repeated Kummer's transformation, — the possibility of which was already indi- cated in the examples 153, 2a (cf. Ex. 130). ea As an example, we may take the series >} 7 which is practically useless 156. k=1 2 for the direct evaluation of its sum ® Here we think of the Ot row and column as consisting aan of noughts, which we do not write down. The choice of the series >’- Shh a for the first column, which was already used 1) on p. 247, then appears obvious enough. This gives 1 2 a, y=Yrein As second column, we shall then, as in 158, 1, choose the series 1 2rEIn GT’ and so on. The k* row of the array thus takes the form 1 0! 1! 21 == 4 = ove, E> k(k+1) kGE+1)EF2) EEL) (+2) (EES) ~ The further calculations are, however, simplified by breaking off this series at the (k—1)* term and adding as k* term the missing remainder 7, after 8 This example is taken from the work of A. A. Markoff: “Differenzen- rechnung”, Leipzig, p. 184, 1896. 266 Chapter VIII. Closed and numerical expressions fot the sums of series. which the series is regarded as consisting entirely of noughts. The kt row now has the form: il 0! 1! ( —2)! ESI ney ey ash Subtracting the terms of the right hand side from the left in succession, we easily find +75. b *&—1) BEET D. Gr-1) ; : Ha In our case, the process of splitting up the series 253 into an array of the form (A) of p. 241 thus gives: Ft 1 1. 0 1! WT. 13 tF3 1 01 1 21 B= 34 is: Hyun TE £m 1! ar (k—2)! (—1! BRIE Bee EY EES EEA Go] Since all the terms of this array are => (0, the main rearrangement . : fo theorem 90 itself shows that we may sum in columns and must obtain 5 as ultimate result. Now in the nt column we have the series 1 1 ] 1! iE i ; tel) Testers 7 Le dtxed) By 132,3 for a =n+1 and p =m, the series in the square brackets has the sum A n(n+1). 2a Hence the ntt column has for sum n ! : i=l =O)! Iver rg rrr (n— 1)! (r—1' SS TyTOR” 2n) Therefore we have z (n=? 3 wi A r=11 2 n=1 (2 n)! This formula is significant not only for numerical purposes, in view of the appreciable increase in the rapidity of the convergence, but almost more so because it provides a new means of obtaining the closed expression for the sum of the series 253 , which we only succeeded in determining indirectly by using the expansion in partial fractions as well as the series expansion of the function cot. In fact we can easily establish directly (cf. Ex. 123), that 121 implies the expansion; for ne =r ls (sin—1a)2=— i tt 2). Exercises on Chapter VIIL. 267 Putting «= 3 , we at once deduce Z | Shea Ol a =32(Z Y= 2, A further application of fundamental importance of Markoff's transforma- tion we have already come across (v. 144) in Euler's transformation, which was indeed deduced from Markoff’s. For further applications of Markoff’s transformation we must refer to the accounts of Markoff himself (v. p. 265, footnote 39) and of E. Fabry (Théorie des séries a termes constants, Paris 1910). Their success depends for the most part on special artifices, but they are sometimes surprisingly effective. Nu- merous examples will be found, completely worked out, in the writings re- ferred to. Exercises on Chapter VIII. I. Direct formation of the sequence of partial sums. 2 22 4 x4 8x8 z 100, 3) Pa Ll TLR TI for lull. ?_ for joi] x x? x4 x8 1-2 2 b) T= rar Iau r 1 for lw] >1. 1 ee on DHE > 2 UT dL). (IT) gent. When does the series still continue to converge for arbitrary a,, and what is its sum? is, for a, positive, ¢nvariably conver- 102. a) Silat — 4 1 1 2 int: -1 - —-1 = —=1 2 (nine: tan a= tan Pd tan i) 2 1 F/1 —1 sss DET ETT 1 ! 1:0 103. a) TTT TINT if +0, —1, =1Fy =x I @ x (z+ 1) 1 : b = annie Ee ———— oe i . ) HIRING Int J Elvoenl a(@+1) a(a+1)(a+2) Lo ob=1 g 142 Irn Tt IeE nea TT HD >ut1>1, 1 k=-b1 (,—b)(F,—b) 1 me 104. Et Ty Db if v0, every ky>0 and 3) & is divergent. v 40 Cf. 2 note by I. Schur and the author: “Uber die Herleitung der Glei- 2 chung Sn=%" Archiv der Mathematik und Physik, Ser. 3, Vol. 26, p. 174. 1918. 268 Chapter VIII. Closed and numerical expressions for the sums of series, i | F 4 4 105. a) 2 ga tan gag =—; n=0 Zid x b) 3 tan —=_—cotz — 2 ? (hint: coty —tany =2cot2y). \- g(n) 5 106. In Yet ,,25,, DETR FET bn ERR CP oh denote fixed given natural numbers, all different, and ¢ +0, —1, —2, ... any real number, while g (z) denotes an integral rational function (polynomial) of degree (rt) = T,, then Bs, (nal trad al 102, n? nl 112. If we write n=1 gp, (2=12,...), then the numbers g, are integers obtainable by the symbolical formula g?tl (1+g?. We have g, = 1 ga =2, Bg =1000n. 1:1 1 113. Sr I aay 1 ~ may be summed in the form of a closed expression by means of elementary functions when z/y is a rational number. 1 1 1 l~rtvrpi-x* 1 1] 1 1 SIE 1 1 1 l=mt5=ftT:"'= +5 - 3 (Fg 1os2), L_(z+2l0g (+1). Special cases are: 2 +1og 2), iy? 10 270 Chapter VIII. Closed and numerical expressions for the sums of series. 114, Writing Sem =L@, (lz]| <1), we have, if (z,) denotes y=l = = Fibonacct's sequence 6,7, oi 3 — (= 3 gimddofes 50-04) i=1 a ® © _ 1)k—1. And if we write >’ 3 =s, and mh we have on ER F=L Ti -1 Tok s III. Exercises on Euler's transformation. 115. We have (for what values of xz?) (=D) 2 k! oY. . 5) 2 z+ n = So zx+1)...(x+k)’ -h. 1 1 x 1.2 #7 \2 9 ~ mr on vols =D = © Zeer mrp) Pl; 116. If we put Dep E ie Se 2 no? n= n=0 z 1 Yd we have b, =A4"a, In particular, therefore, at? 2 (el2)(0fd)a’ ...] TI "Gi DE 03) 1 of + 1 at a+12t.1! hea Dz. 2 — 1)2-1 o£ Sn -3n2 (n= get). n=1 % a) e aly pap ott =14 117. Quite special cases are: a) D-20) +6) =. eal Tam lt frat 1/n 1 /n 1 n 2.4... (@n (— 1)" a, satisfies the inequalities : n=0 § ll la, 406 2 pt Dred lio ial 2480.04 Use this to prove the Wh 3 1 x x? 1 lim is — .]= : eriotd ITT 11a ® Exercises on Chapter VIII, 271 120. If s; and S, denote the partial sums of both the series in 144, we have n1 n-+1 n+ 1 Cid Jatt (71) Sp = on+1 Use this relation to prove the validity of Euler's transformation. 121. The following relations hold, if the summation on either side is taken to start with the index 0 and the difference-symbols 4 operate on the coefficients on the left, ay, a,x, a,;, respectively: a) J(—Deaqpa*=(1—-y) ZA" a,-y* with (14+2)(L—») =1; b) T(— Day a¥=(1—y?) ZT 4%a,-y®* with 1+22)(1—y%)=1; ©) Z(—Dkay et =y1—y2. ZT A%a, p22 +1 with (1 +2?) (1 — 99) = 1. 122. Thus e. g. diy on | 2: ad 22 x2 ) 7 tan Yr = - | 14 T5 + : 1422 1 Putting x = L L l this provides peculiarly convenient g BU 3 ) wy 91 79’ wieiey p p y series for mz, as for instance x el Gols sd 2/92 3 2/1 Fowmri bimodal 2D, "= eril wl TES ad 7 = 9 tan 5 4 tan " = 5 tan 7 + 8 tan 7g and others. 123. The preceding series for tan—! x may also be put in the form sinTty yi=+ Hence deduce the expansion ae 5 coe, y+ 2p ps cy = - 5 B-1l° 9 1 on, (sin)? = 2 TA 29) IV. Other transformations of series. 124. Writing - = S,, we have n=2" 3 a) Sp+ Sp +S, +---=1; b) Sr St Sete =p 1 1 1 ©) STS =r d) Sut Sut-5 St: =log2 Le % & 5,~5+5,—+. = pe 1 1 7 i DS —5S+gS—+:=log p— ro = 1 1 g) v5trS +551 a l-C; h) 15, =1s +—.ve=log2+-C—-1, where C denotes Euler's constant, defined in 128,2 and Ex. 86 a. J i / | : $4 979 Chapter VIII. Closed and numerical expressions for the sums of series. bi cdl 125. With the same meaning for S, as in the preceding exercise, writing 1 1 +s Ale” (r3g—77) =* and lim ih we have by Sy +b; S;+-+-=1—logi. (The existence of the limit 1 results from the convergence of the series. We have A=v2m.) 126. 1 . ! 17 1 1 2) 2 CIOL EIT irra a” b SL a aw te? — wee, ) 231 142° a?” ie > 1 1! 1 2! Hi 1 127. ein TE Le . 3 vy Toe lD Sieg’ AIT 1 b) =z BDI + 2° = 10 p= a2. 128. With reference to § 35 A, establish the relation between = il z 1 d a SE LS Saree iarF Grats 2 GFe-1...@retor] and, by giving special values to « and p, prove the following transformations: Zl 9 S 1 D SWF TEE LR Lim neg 1 : STEHT 5 SL Water +3 nt ap nS la-ln 23 ! : — (m+1)2 2520 35 = WF (n+ 1)% (n+ 2)% (n+ 3)3 Evaluate the sum of the first series to 6 places of decimals. 129. Prove similarly the transformations: Tnm+ 142 5 1 5 2) = = Gus (nt 1p % £1 7 s 28n2(n+3)+24nt5 8 SPE=T 2, 12#8 (n+ 1)® : 00 (=z? ©) = m+)... 4p -1)7° cSp32 1 plo 1 § (- 1) 4(p+1) (YH? 4 ES nmtlp. (nrp+1p Evaluate the sums of the series a) and b) to 6 places of decimals. Exercises on Chapter VIII. 273 130. a) Denoting by T, the sum of the series c) in the preceding exer- cise, we obtain relations between IT, and Tez, 7; and T2z+1. What are these relations? Is the process £2 — 00 allowed in them? What is the transformation thus obtained? Is it possible to deduce it directly as a Markoff transformation? b) We have log 2 = Si n=1 Er {-D=—? 8 1 " 3+ Pig aD io Give the form now taken by the sh ntion of the series for log 2 which were indicated in a). c) Carry out the same process with the series 122 for a 4 1 (og, ny’ where » starts from 131. The sum of the series TEL 2 the first integer satisfying log, » >> 1, evaluated to 8 decimal places, is exactly ~ 1:00000000. — How may we determine whether the actual decimal expan- sion begins with 0-... or with 1-...? — The solution of this problem requires a knowledge of the numerical value of ¢’' = ¢¢) to one decimal place at least; this is ¢// =3814279.1... It suffices, however, to know that ¢”/ —[¢"'] = 0-1... (Cf. remarks on p. 249.) 132. Arrange in order of magnitude all natural numbers of the form p?, (p, q positive integers => 2) and denote the nth of the numbers so arranged by p,, so that : (P Par 0 = 29, 16,25, 27,32, ...). We then have a 2 Pa—1 (CL. 68,5.) 158. Pare HY Development of the theory. Chapter IX. Series of positive terms. § 36. Detailed study of the two comparison tests. In the preceding chapters we contented ourselves with setting forth the fundamental facts of the theory of infinite series. Henceforth we shall aim somewhat further, and endeavour to penetrate deeper into the theory and proceed to give more extensive applications. For this purpose we first resume the considerations stated from a quite elementary standpoint in Chapters III and IV. We begin by examin- ing in greater detail the two comparison tests of the first and second kinds (72 and ¢3), which were deduced immediately from the first main criterion (¥0), for the convergence or divergence of series of positive terms. These, and all related criteria, will in the sequel be expressed more concisely by using the notation X¢, and 2d, to de- note any series of positive terms known a priori to be convergent and divergent respectively, whereas 2'a, shall denote a series — also, tn the present chapter, of positive terms only — whose con- vergence or divergence is being examined. The criterion '¥2 can then be written in the simple form I nS Cu : ec, Un =n : 2. This indicates that, if the terms of the series under consideration satisfy the first inequality from and after a certain n, then the series will converge; if, on the other hand, they satisfy the second inequality, from and after a certain mn, then it must diverge. The criterion '¥83 becomes in the same abbreviated notation a c a d (In) n41 < n+41 . e, n41 > n 41 : QD. a £22 0, « Tem od n n n n Before proceeding we may make a few remarks in this connexion. But let us first insist once more on one point: Neither these nor any 3 § 36. Detailed study of the two comparison tests. 275 of the analogous criteria to be established below will necessarily solve the question of convergence or divergence of any particular given series. They represent sufficient conditions only and may therefore very well fail in special cases. Their success will depend on the choice of the comparison series X'¢, and 2d, (see below). The fol- lowing pages will accordingly be devoted to establishing tests, as numerous and as efficacious as possible, so as to increase the pro- bability of actually solving the problem in given special cases. Remarks on the first comparison test (157). 159. 1. Since for every positive number g the series Jgc, and Xgd, necessarily converge and diverge respectively with 2'¢ and 2'd,, the first of our criteria may also be expressed in the form: Sge(cta) + 8 Tae: B® or, even more forcibly, in the form lim 2 < + co 2 ec, lim 2% > 0 : 2D, 2. Accordingly we must always have: oe o LG 7 lim Tr + co, lim n= 0 or, otherwise expressed: T= a : i 2 1 lim —* = 00 is a mecessary condition for the divergence of 3 a,s n . a lim" =0 nn mecessary ” » nn convergence » 2a. SR 3. Here, as in all that follows, it is not necessary that actual unique limits should exist. This may be inferred, to take the question quite generally, from the fact that the convergence or divergence of a series of positive terms remains unaltered when the series is sub- jected to an arbitrary rearrangement (v. 88). The latter can in every case be so chosen that the above limits do not exist. For instance 2c, can be takento be 14-1 +141 -+---, and Za, to be the series 1 x 1 1 ditt bds tian, obtained from the former by interchanging the terms in each successive n pair; the ratio > certainly tends to no unique limit; in fact, it has n distinct upper and lower limits 2 and i. Similarly, let 3d, be chosen to be the series 1 +3 +3431 +..., nd let 2a, be the series / sob beh de dled 4d fogs if pels 276 Chapter IX. Series of positive terms. deduced from the former by rearrangement, (in this series every two odd denominators are followed by ome even one). Here oe has the n two distinct upper and lower limits 2 and 2. In a similar manner we may convince ourselves by examples in the other cases that an actual unique limit need not exist. If, however, such an unique limit does. exist, it necessarily satisfies the conditions indicated for lim and lim, since it is then equal to both. 4. In particular: No condition of the form ren 0 is necessary n for the convergence of Xa, — unless all the terms of the divergent series 2'd, remain greater than a fixed positive §. For, even if we only have limd, = 0, by choosing RBi Sat ie (%) is a monotone descending sequence, whose limit n n+1 n y is defined and > 0. In particular lim 2 = y < -}- 00, and, by 159,1, 2a, is convergent. In the case marked (9), 2 is monotone ascend- ing from and after a particular #, and accordingly also tends to a definite limit > 0, or to -- co. In either case the condition lim o >0 Se — n of 159, 1 is fulfilled and this shows that Yq is divergent. 2. The comparison test II thus appears as an almost immediate corollary to the comparison test I. If the convergence or divergence of a series Xa, can be inferred by comparison with a (definitely chosen) series X'¢, or 2d, in accordance with 158, then this may also be inferred by means of 157 (or 189, 1), but not conversely, 1. e. if I is decisive, II need not be so. Examples of this have already occurred in the pairs of series of = 159, 3. For the first pair we have fim 22 = 2, while fon alter- n n § 86. Detailed study of the two comparison tests. 277 1. iy : ; nately = 2 and == sole it is sometimes greater, sometimes less than the : ae ; : 1 corresponding ratio 2d, since this constantly = 5 The second n pair of series represents an equally simple case. 3. This relation between the two types of comparison tests be- comes particularly interesting when we come to deal with the two tests to which we were led in § 13 as immediate applications of the first and second comparison tests. These were the root and ratio tests, inferred from I and II by the use of the geometric series as com parison series, and they may be stated thus: — i p, we always have n,— 7” — & Ve, < VA (iW +3). n— & n — & : But V4 —1, and hence (w : a Yd w+ 5 We can therefore n_— so choose #n, > p that, for every n > n,, we have (w -+- 3) VA n,, n Va, =0 or lim == =i-|-00, n If the limit of this ratio exists and has a finite positive value, or if it be known merely that its lower limit > 0 and its upper limit < -} 00, then the convergence of the two series will be said to be of the same kind. In any other case a comparison of the rapidity of convergence of the two series is. impracticable?®. Definition 2. If Xd, and 2d are two divergent series of posi- tive terms, whose partial sums are demoted by s, and s,' respectively, the second is said to diverge move or less rapidly (or more or less markedly) than the first according as oa hy lim=* = 400 or lim = =0. Sn Sn If the upper and lower limits of this ratio are finite and positive, then the divergence of both series will be said to be of the same kind. In any other case we shall not compare the two series in respect of rapidity of divergence?. The two following theorems show that the rapidity of the con- vergence or divergence of two series may frequently be recognised from the terms themselves (without reference to partial sums or re mainders): Theorem 1. If Ce (+00), then Zc, converges more (less) rapidly than 2c,. ’ sy # In the case lim - =0 (> 0) and Tim on < + 00 (= + 00), we might also TT n speak of the series Yc,’ as “no less” (“no more”) rapidly convergent than the series 2 ¢,; this however presents no particular advantages. In the case of the lower limit being 0 and the upper limit 4 00, the rapidity of the conver- gence of the two series is totally incommensurable. A similar remark holds for divergence. (The student should illustrate by examples the fact that all the cases mentioned can really occur.) — These definitions may be directly transferred to the case of series of arbitrary terms, replacing 7, and 7,’ by their absolute values. ; 4 The properties referred to in these definitions are obviously transitive, i.e if a first given series converges more rapidly than a second, and this again more rapidly than a third, the first series will also converge more ra- ~ pidly than the third. ’ 280 Chapter IX. Series of positive terms. Proof. In the first case, given ¢ we choose 7, so that for every n> n, we have ¢’ 3 I= 1 3» “(Esa 9 (3) A. a2" 2 4 7 Bea 1.2.3 log which tends to 0. Similarly Fao (by 38,6); the other cases are even simpler. 2. The series 1 1 1 JE, Sno Sls 2% Duta 2 nlog n log, n’ are such that each diverges less rapidly than the preceding. Besides the above simple examples, the most important cases of series with rapidity of convergence forming a graduated scale are afforded by the series which we came across in § 14. As we saw in that Ey the series | 3 3 1 1 *s a n)® “ nlogn (log, n)* >: hn 2 log n...log,_, n-(log, n)* converge for ¢ > 1 and diverge for ¢ <1. Our theorems 1 and 2 now | show more precisely that when p is fixed each of these series will converge or diverge less and less rapidly as the exponent « approaches ; unity (remaining > 1 in the first case and <1 in the second). Simi- 1 larly each of these series will converge or diverge less and less ra- § 37. The logarithmic scales. 281 pidly, as p increases, whatever positive® value may be given to the ex- ponent « (> 1 in the first case, <1 in the second). The second alone of these statements perhaps requires some justi- fication. Divide the generic term of the (p +4 1)! series with the ex- ponent ¢/ by the corresponding term of the pt series taken with the exponent ¢. We obtain (log, n)* log, n-(log,, 4 n)* 7. In the case of divergent series, ¢ and ¢ are positive and <1; the ratio therefore tends to 0, q. e. d. In the case of convergence, i.e. « and « both > 1, the ratio tends to 4 oo; in fact, — by reasoning analogous to that of 88, 6, — we have the auxiliary theorem that the numbers (logp +1 ny” {log (log, m) (og, n)f Sk (log, n)f form a null sequence, f= « — 1 denoting any positive exponent and p any positive integer. This proves all that was required. The gradation in the rapidity of the convergence and divergence of these series enables us to deduce complete scales of convergence and divergence tests by introducing these series as comparison series in the tests I and II (p. 274). We first immediately obtain the fol- lowing form of the criteria: 0 ig 1 a fe : e 2,2 nlogn...log, , n(logpn) a1 : 9 164. (In) a, iy = wos log » on logp—1n : ( log, n ¥ a, > n+ 1 log (n+1) log,_, (#41) \log,n +1) o>1 s ce with <1 : 9. These criteria will be referred to briefly as the logarithmic tests of the first and second kinds — also in the case p =0. Their effi- ciency may be increased by the choice of p, and, for fixed p, by the choice of «, in accordance with our previous remarks®. 5 For « =— f <0, each series of course diverges more rapidly than the \B a preceding one with the exponent replaced by 1; thus e. g. og w , with n #> 0, diverges more rapidly than 22. 8 The convergence and divergence of series of the above type was known to N. H. Abel in 1827, but was not published by him (CEuvres II, p. 200). A. de Morgan (The differential and integral calculus, London 1842) was the first ei Chapter IX. Series of positive terms. For practical purposes it is advantageous to give other forms of these criteria. Such transformations are given below with a few remarks appended, but without completely carrying out the necessary calculations. Transformation of the logarithmic tests 164, I. 1. When a and b are positive, the two inequalities a 0 : 9D. a test of practically the same effect. The parts relating to convergence are indeed completely equivalent in (I’) and (I”); that relating to divergence is not quite so powerful in (I”) as in (I), since it is required in (I”) that 4, should remain, from some value of # onwards, not merely = (0 but greater than a fixed positive number?. 3. If we use the somewhat more explicit notation dom AD and consider both AP and 4210, we obviously have log, n 4210 1.1 2p LAD ® log, un 8 log, n log, n 2 log, ., n log (log, n) And since, by 38,6 tends as # increases to + 00, this simple transformation leads to the following result: If for a particular » one of the limits of A= AD is different from zero, it is necessarily + co for the following p. More precisely, if we denote by up, and x, the upper and the lower limits of 4, =47, for every p, then if we have, for any particular p, xp Sup<<0, wehave x,4,=py4;=—00 and if He 4,0, ‘wehave . pu =wuyt,=+00. If, however, #0, 1, >>0, we have 4,,,=-00, Uris =00- The scales of reference (I) thus lead to the solution of the question of conver- gence or divergence if, and only if, for a particular p, the values #, and u, have the same sign. If the sign is negative, the series converges; if positive, to use these series for the construction of criteria. Essentially, these criteria are consequences of 164, I and 1I; numerous transformations of them were subsequently published as special criteria, e. g. by J. Bertrand (J. de math. pures et appl, (1) Vol. 7, p. 85. 1842), O. Bonnet (ibid., (1) Vol.8, p. 78. 1843), U. Dini (Giornale di matematiche, Vol. 6, p. 166. 1868). ? It would clearly, however, be wrong to write the last D-test in the from lim 4, => 0, since the lower limit may very well be 0 without a single term being positive. § 37. The logarithmic scales. 283 it diverges; if the two numbers have opposite signs for some value of p, then for all higher p's we have ; ) Tim 42 = Im AP =—o0, Iim4P =+00 and the scale therefore is not decisive. Similarly it fails when both numbers are zero for every p. Transformation of the logarithmic tests 164, II. 166. 1. The following Lemmas are easily proved: Lemma 1. For every integer p=0, for every real o and every sufficiently large nm, an equality of the form log, (n — ny. 1 o Olea, ( log, n - nlogn... log, #2’ holds, where (9) is bounded®. We immediately infer that, for every integer p = 0, for every real « and every sufficiently large =, : n— 1 log (n—1) : log, (n— D. (8 (n— De n log» logy log, n 1 1 1 1 a a Ta "nn nln nlogn...log, _,n mnlogn...log,n n? where (7,) is again certainly bounded?. Lemma 2. Let Ja, and Xa, be two series of positive terms; if the series 167. whose n't term is ay, m={ An he 1s absolutely convergent, the two given series ave either both convergent ov both divergent. In fact, we have y,>—1 for every »; taking, then, any positive in- teger m, writing down the relations yyy 5a = eT + Ps» for y=m, m+1, ..., n—1, and multiplying them together, we at once de- duce that the ratio a,’ a, for n> m lies between two fixed positive numbers, — 8 An equality of the above form of course holds under any circumstances. In fact we can consider the numbers 9, as defined precisely by the equation: =n 1 — o — (8 ii Ly] : nlogn...log,n log, n The emphasis lies on the statement that (J,) is bounded. — The proof is ob- tained inductively, with the help of the two remarks that if (4,/) and (9,”) are defined, for every sufficiently large n, by ” l-2)*=1—0ax,— 9, 2,2 and tog (1 1 ) 1 On Ral PA Vn HE they are necessarily bounded, provided (x,) is a null sequence and the num- bers y, are in absolute value => 1, say. ® The interpretation in the case p = 0 is immediately obvious. 284 Chapter IX. Series of positive terms. n+1 The conditions of the Lemma are fulfilled, in particular, when the ratios % | Ant1 anit an a, ’ a . . Yh z and = 1 lie between fixed positive bounds and the series > n converges. 2. In accordance with the above we may express the logarithmic test the second kind e. g. in the following form: = ’ ant1 > 1 1 1 « 168. “Eh —— — - : a, = n nlogn nlogn...log, —g lt ~ nlogn...log,n 3 a>1 : eC > W1 ae. <1 : D, or, after a simple transformation, wd Liar 2 169. [ute Si —1+- mp + EET) nlogn ...log,n S=48<0 : e =0 : PD, or, finally, denoting the expression on the left hand side for brevity by B,, and slightly restricting the scope of the D-test (cf. 165, 2), lim B, < 0 : g, limp ~ 0 : 9D. Remarks analogous to those of 163, 2 hold here. 3. The developments of 165, 3 also remain valid, with quite unessential alterations. For, if we use the more explicit notation B, = BY), we have ob- viously BPD _14 BP.log,,, n And, as log, ,n— +00, we may reason with this relation in precisely the same manner as with its analogue in 1685, 3. It is unnecessary to develop this in detail. 4. Still more generally, we may at once prove that a series of the form 1 > (a —1 e I. 1% (log n)*1 (log, )%... (log, n)* converges if, and only if, the first of the exponents «, og, cy, + «+, co, which differs from 1 is >> 1. The values of the subsequent exponents have no further influence. — When the comparison series is put into this form, Raabe's test (§ 88) and Cauchy's ratio test appear naturally as the 0% and the (—1)® terms of the logarithmic scale. § 38. Special tests of the second kind. The logarithmic tests deduced in the preceding article are un- doubtedly of greater theoretical than practical interest. They afford in- deed a more profound insight into the systematic theory of the con- vergence of series of positive terms, but are of little use in actually testing the convergence of such series as occur in applications of the 10 Here we make the #'® term of the investigated series Xa, correspond to the (n—1)'™ ferm of the comparison series, which, by 82, theorem 4, is allowable. 8 38. Special tests of the second kind. 285 theory. (For this reason we have only sketched the considerations relating to them.) For practical purposes the first two or three terms, at most, of the logarithmic scales may be turned to account; from these we proceed to deduce by specialization a number of simpler tests, which were discovered at various times, rather by chance, and each proved in its own way, but which may now be arranged in closer connexion with one another. For p = 0 the logarithmic scale provides a criterion already estab- lished by J. L. Raabe. We deduce it from 169, first in the form [ate pv 1s —5<0 : c Gn % 0 : 9, or, as we may now write more advantageously, pti Sel << wm] : c [ —]n {27} oS 170. n nn The very elementary nature and great practical utility of this cri- terion makes it worth while to give a direct proof of its validity: the @-condition means that, for every sufficiently large #, IV IA tly 2 or na, Smn—1a,— pa, n where S =a — 10. Hence w—Ne —na pe, >0 and therefore na, ,, is the term of a monotone descending sequence, for a sufficiently large nn. Since it is constantly positive, it tends to a limit y > 0. The series X¢, with ¢,=(n — 1)a, — na, ,, there fore converges, by 181. Since 4 < 5 x = 70, the: convergence: of 2a, immediately follows. — Similarly, if the 9-condition is fulfilled, we have 2 1 Stiot-L oo w-1e,—na,, <0. Accordingly na, ,, is the term of a monotone increasing sequence and therefore remains greater than a fixed positive number yp. As a 0, > z, y > 0, the divergence follows immediately. If the expression on the left in 170 tends, when n— co, to a limit /, it follows from the reasoning already repeatedly applied (v. 76,2) that / << — 1 involves the convergence of 3 aad’ > —, its divergence, while / = — 1 leads to no immediate conclusion. 1 Zeitschr. f. Phys. u, Math, von Baumgarten u. Ettinghausen, Vol. 10, p. 63, 1832. Cf. Duhamel, J. M. C.: Journ. de math. pures et appl., (1) Vol. 4, p. 214, 1839. 286 Chapter IX. Series of positive terms. Examples. 1. In § 25 we examined the binomial series and were unable to decide there whether the series converged or not at the endpoints of the interval of convergence, that is, whether for given real o's the series SY) sat Sel n=0 \" n=0 os 4 were, or were not, convergent. We are now able to decide this question. For the second series we have ppg O@—0 th—(etDh a, wil n+ 1 Since this ratio is positive from a certain stage on, it follows that the terms then maintain one sign; this we may assume to be the sign +, since changing the signs of all the terms does not, of course, affect the argument. Further, ac- cording to this, : (Seer tht 5 ay, Gn nl from which we at once deduce, by Fnie's test, that the second of our series converges for ¢ > 0, and diverges for « <0. For «= 0, the series reduces to its initial term 1. > ~ For the first series we have Stil yo 05] a, ntl and, since this value becomes negative from some stage on, the terms of the series have an alternating sign from that stage on. If now we suppose «410, we therefore have Ani ay 2 whence we infer that ultimately the terms a, are non-decreasing. The series must therefore diverge. If however we suppose «+ 1 > 0, we have ultimately, say for every un =m, at1 (a) n+ 1 and the terms ultimately decrease in absolute value. By Leibniz's criterion for series with alternately positive and negative terms, our series must therefore Ay +1 an =1-— = ¥, converge, provided we can show that (5) 0. If we write. down the rela- tions (a) for m, m4 1,...,n—1 and multiply them all together, we deduce for every n >m n 3 an] = lan] J (1-225). | smd v at1 v Since, however, the product [] (1 — ) , by 126, 2, 3, diverges to 0, a, must o . also — 0, and therefore >’ (5) must converge. Summing up, we therefore have the following results relating to the binomial series: The series > i x" converges if, and only if, either |x| <<1, or z=—1 n=0 § 38. Special tests of the second kind. 287 and o> 012, or x=-+1 and «> —1. The sum of the sevies is then by Abel's theorem of limits always (14+ 2)*. In all other cases the sevies is divergent. (An appreciable addition to this theorem is provided by 247.) 9. The following criterion does not differ essentially from that of Raabe; it is due to O. Schlomilch: Ay t1 S—a<—1 : c nlog~—" { 1 : 9. n In fact, in the case (9), we have, by 114, + 1 An +1 Tn ay > = = ce. 2 >1 3 (A 3 S i from which the divergence follows by Raabe's test. In case (€) we have, ultimately, a Tila, Bl o a, = n ’ I if «>a’">1. By 170, this involves convergence. If, in the logarithmic scale, we choose p =1, we obtain a cri- terion of the second kind which, omitting the limiting case « =1, we may write Yt 1 > %p . On = a>1 : e sax] : 9. A direct proof of the validity of this criterion can be given as follows: As in the proof of Raabe’s test we first put the criterion in the following form: 2 ipa, with 3>0.r.€ —1+4+ (mn —1)lognla, — [nlogn]a = 5 + Hog ula, ep log) dS with f/>0-: 9. If now the C-condition is fulfilled, since, as we may immediately verify by 114, «, (n—1)log(n —1) > — 1+ (n — 1)logn, we have a fortiori (n 253 1)log (n i 1)-a, = nlogn-a, = pa, Accordingly nlogn-a,,, is the term of a monotone descending se- quence and accordingly tends to a limit y > 0. By 131, the series whose nth term is ¢c,=(n—1)log(n —1)-a, —nlogn-a,,, If, on the other hand, the D-condition is fulfilled, we have (mn — 1)log(n — 1)-a, — nlogn-a,, 1 For n— J co, however, the expression in square brackets -— — fp’ 1 3 must converge. As q, < 7 Cn the same is true-of Ja. 2 For «=0, see ahove. 988 Chapter 1X. Series of positive terms. (by 112, b), and is therefore negative for every sufficiently large =. Hence for those n’s the expression nlogn-a,,, increases monotonely and consequently remains greater than a certain positive number y. Asa... = TI y > 0, it follows that Xa, must diverge. Here again we may observe, as repeatedly in previous instances, that, if ¢, tends to a limit /, then }>1 involves convergence, and l <1 involves divergence, while, from ! = 1, nothing can be directly inferred. Even this, the first properly logarithmic criterion of the scale, will rarely be actually applied in practice. In fact, the series which are amenable to this test, and not already to a simpler one (Raabe’s test, or the ratio test), occur exceedingly seldom; and as their convergence is no more rapid than that of sl = (e > 1), these series are n (log n)® useless for numerical calculation. It enables us, however, to deduce easily one or two other cri- teria. We will above all mention 172. Gauss’s Test3: If the ratio bh can be expressed in the form An +1 wr] = aT % An ” n” where 2 > 1, and (9,) is bounded®, then Xa, converges when o>1 and diverges when a <1. The proof is immediate: when ¢>=1, Raabe’s test itself proves the validity of the assertion. For a« =1, we write Zitr ogo Xo 1 (ue), a, ne ulognl\ ,4-t [2 and as now the factor in brackets tends to zero since (1 — 1)> 0, the series certainly diverges, by 171. ; Gauss expressed this criterion in somewhat more special form as follows: “If the ratio oS ti can be expressed in the form n Ay py Bp hyn. . Lb, = Ih Ln (k an integer >1) then Za, will converge when by —b,' < — 1 and diverge when b, — 1b,’ > — 1.” — The proof is obvious from the preceding. 13 Werke, Vol. 3, p. 140. — This criterion was established by Gauss 1 in 1812. : . 14 Cf. footnote 8, p. 283. § 88. Special tests of the second kind. 289 Examples. 1. Gauss established this test in order to determine the convergence of the so-called hypergeometric series wf a@tD) BB+D , e@tDe+? BEFDE+D ,, TTT ART Ter ihe = Seth @ha=D f+). Ghn-1) , a 1.2...» y+... +n-0 where «, 8, y are any real numbers different from 0, —1, —2, ...15, Here Guy _ (etm) (B+) = A EDG+Y which shows in the first instance that the series converges (absolutely) for jz| <1, and diverges for |2|>1. Accordingly it only remains to examine the values z=1 and x =—1. This is analogous to the case of the binomial series, to which, of course, the present one reduces when we choose f =p (=1) and replace ¢ and by —« and — x. For z=1, we have nyt _ nt (at pnt ap an "+@+Dnty ’ This shows that for every sufficiently large %, the terms of the series have one and the same sign, which may be assumed positive. Gauss’s test now shows that the series converges for a+f—y—1<—1, i.e. for «+p 1 converge when o+f—yp <1. We have only to verify further that it also diverges when of —y=1, as this does not follow from precisely the same reasoning as before. If for every n>p > 1 we have - fats = (1 +22) with |9,| <9 for every n, n then, assuming p chosen so large that p% > 9, i101 1-2) (1-2) (1-52) Since on the right hand side we have the product of the first (» — p) factors of a convergent infinite product of positive factors, it follows that |a,|, for all these values of nu, remains greater than a certain positive number. The series can therefore only diverge. 15 For these values, the series would terminate or become meaningless. For n = 0, the general term of the series should be equated to 1. 18 As before, (J,) denotes a bounded sequence of numbers. 173. 290 Chapter IX. Series of positive terms. 2. Raabe’s C-test {ails if the numbers «, in the expression Ing 3..5 an n though constantly > 1, have the value 1 for lower limit. In that case, writing ap, = 1+ f,, the condition is a mecessary condition for the convergence of Xa,” In fact, if nf, were bounded, we should have Garay. 1 2 a, nn 7: and 3a, would be divergent by Gauss’s test. § 39. Theorems of Abel, Dini and Pringsheim and their application to a fresh deduction of the logarithmic scale of comparison tests. Our previous manner of deducing the logarithmic tests invests these, the most general criteria yet obtained, with something of a for- tuitous character. In fact everything turned on the use, as comparison series, of Abel's series, which were obtained themselves only as chance applications of Cauchy's condensation test. This character of fortuitous- ness disappears to some extent if we approach the subject from a different direction, involving a greater degree of inevitableness. Our starting point for this is the following wo | Theorem of Abel and Dini'®: If Xd is an arbitrary divergent n=1 series of positive terms, and D, = d, +d, + --- +d, denotes its partial snms, the series converges when « > 1 0 © d SO = n=1 n=1D, | diverges when a <1. Proof. In the case ac =1, otro it rt oy Topp i lay oy 77 10, 3 Dptq Dui - Dy 1x ’ D, 1x As D,— co by hypothesis, we can therefore choose k =#% each n, so that 5, “ . 1. Fs for § n? Dy stn 1? Cahen, E.: Nouv. Annales de Math., (3) Vol. 5, p. 535. 3 18 N. H. Abel (J. f. d. reine u. angew. Math, Vol. 3, p. 81. 1828) only ° “_; U. Dini (Sulle serie a termini positivi, An- 3 nali Univ. Toscana Vol. 9. 1867) established the theorem in the above com- plete form. It was not till 1881 that writings of Abel were discovered ((Euvres II, p. 197) which also contain the part relative to convergence of the theorem given above. ; § 39. Theorems of Abel, Dini and Pringsheim. 201 by 81, 2, the series 2'a, must accordingly diverge when « =1, and a fortiors when « <1. The proof of its convergence in the case « > 1 is slightly more troublesome. We may at the same time prove the following extension, due to Pringsheim™®. Theorem of Pringsheim: The series 174. = dy, mit 0: Dnu = ETT n=2 D,-Df_, n=2 Dp D2 4 where d, and D, have the same meaning as before, converges for every ¢ > 0. Proof. Choose a natural number $ such that 2 Since, further, the series > 1 1 2 FT or) n=2 DA D, converges, by 131, since D,_, < D, — + oo, and since its terms are all positive, it would also suffice to establish the inequality — D7 D, Cy 1 71) or 1- Beg dfs 2 md 5) IT z = Dy-D,_4 ’ Dy_y Dy % D, that is to say, to prove that 1 Dy—1\0 (t—2") 1 nE1 n° | diverges for ax, and the theorem in 2. shows further that in the latter case we have for a =1, qo 1 Lt de (cf. 128, 2). b) Now choosing for Xd,, in the theorem of Abel-Dini, the series 3 newly recognised to be divergent by a), and replacing, as we may by 1.and 2, D, by D,’ =log n, we conclude that 2 1 converges when o>1 n=2 n (log n)* The theorem in 2. shows further that 1 | il Seed Teal “alan diverges when «<1. | ~~ log log n =1log,n. 20 vy, Cesaro, E.: Nouv. Annales de Math., (3) Vol. 9, p. 353. 1890. : 21 This condition is certainly satisfied if the numbers d, remain bounded, — i hence in all the series which will occur in the sequel. § 89. Theorems of Abel, Dini and Pringsheim. 293 c) By repetition of this extremely simple method of inference, we obtain afresh, and quite independently of our previous results: Starting from a suitably large index (e, + 1)*, the series 1 { converges when o> 1, nlogn...log,_, n(log,n)* | diverges when «<1, whatever value is given to the positive integer p. The partial sums of the series for aw =1 satisfy the asymptotic velation La 1 —_ ~~ log, 4,7. rer OEP Tog, milog, en 2 4. A theorem analogous to 178, but starting from a convergent series, is the following: Theorem of Dini. If 3c, is a convergent sevies of positive terms, and Ypn_1="Cn-+Cpnyq,+--- denotes its remainder after the (n— 1) term, then 3 Cp il Cor { converges when o <1, eT, Catonpst 9” diverges when o=1. Proof. The divergent case is again quite easily dealt with, since, for ao=1, Cn doe bi ale Cnt+k ~ treating q Tete, = = ) Yn—1 Yn+k—1 V1 "n—1 and for every (fixed) n, this value may be made >1 by a suitable choice of k, as r,—0. By 81,2 the series must therefore then diverge. For a >1 this will a fortiori also be the case, since 7, is <1 for every sufficiently large nn. alia 1 If, however, ez << 1, we may choose a positive integer p so that oo <1 — —, and it now suffices — again because 7, << 1 for » > n, — to establish the con- vergence of the series Cn Y—1 =n 7 ——— —_— oh i= = 3 pi n—1 where 7 = lL. Now 7, tends monotonely to 0 and consequently Br: y~1)) is cer- tainly convergent with positive terms. It therefore suffices to show that "'h—1—¥n = 1 T v ite C4 eel — : hi ral i] that is to say 1 (1-90) oll -9) Iw % ih Tn—1 But the latter relation is evident, since 0 > 1, if » be taken to start from the value (e, +1). 28 vy, footnote 18, p. 290. 176. 294 Chapter IX. Series of positive terms. § 40. Series of monotonely diminishing terms. Our previous investigations concerned for the most part series of quite arbitrary positive terms. The comparison series used for the construction of our criteria, however, were almost always of a much simpler nature; in particular, their terms decreased monotonely. It is clear that for such series simpler laws altogether will become valid and perhaps also simpler tests of convergence may be constructed. We have already shown, e. g. on p. 124, that if in a convergent series 2c, the terms diminish monotonely to zero, we have neces- sarily nc, — 0, a fact which need not occur in the case of other convergent series (even with positive terms only). Again, Cauchy's condensation test '¢'¢ belongs to the series we are considering. We propose to institute one or two further investigations of this kind and, in the first instance, to deduce for such series a few very simple and at the same time very farreaching criteria. Their con- vergence, as we shall see, is often very much more easily determined than that of more general types of series. 1. The integral test®. Let Xa, be a given series of monotonely n=1 diminishing terms. If there exist a function f(x), positive and monotone decreasing for x > 1, for which =a, for every n, then 2a, converges if, and only if, the numbers n L=Jr@a 1 are bounded *>. Proof. Since, for (k—1) Stk, we have f() >a, and} for k 2), it follows (by § 19, Theorem 20), that k+1 k Jroas,— J >a, — J,>0. — The limit in question is therefore certainly positive, if f(¢) is strictly monotone decreasing. Examples and Illustrations. 1. This test not only enables us to determine the convergence of numerous series, but is also frequently a means of conveniently estimating the rapidity of their convergence or divergence. Thus e.g. we can see at once that for «> 1 the series n A : at 1 1 1 yr; since pi Seskyly nt 1 n=1"7 must converge, whereas n 1 at N —, where J,=| —=logn—+ 00, im ¢ n=1 1 must diverge. But we learn further that, for « > 1, n+k+1 Wn n+k at 1 at emits 2 v=n+1 p* 2 ntl and therefore 1 1 1 1 ——— << — a1 WHIT TE Ty yr For «¢=2, this evaluation was already established on p. 259. In the same way the supplement to 176 gives a fresh proof of the fact that the difference 1 1 [145+ 5 toga] is the term of a monotone descending sequence tending to a positive limit between 0 and 1. This was Euler's constant mentioned in 128, 2. Similarly, the supplement also shows that when 0 << ¢ << 1, the difference 1 ipa Prin ar 1 ~ isthe term of a monotone descending sequence with a positive limit less than 1. Therefore, in particular (cf. 44, 6), for 0 << eo <1: 1 14 2a. toe 1 1 1 nie T= let LN Zit t= Er l—a! and it is easily seen that this relation holds equally when «<0. 177. 296 : Chapter IX. Series of positive terms. 2. More generally, from 1 1 dt = ——, if x+1 —_—e =) 4 (og 21] yy tlogt...log _1t-(log, p > p log, 117, fo=1, we can immediately deduce, by the same method, the known conditions of convergence and divergence of Abel's series. We have now three totally distinct methods of obtaining these. The supplement to 176 again affords us good evaluations of the remainders in the case of convergence, and of the partial sums in the case of divergence. 3. If f(x) be positive for every sufficiently large x, and possesses, for those z's, a differential coefficient equal to a monotone decreasing (also posi- tive) function with the limit 0 at infinity, the ratio f’ (x)/f (x) is also mono- tone decreasing. Since (Os Fy dt =loe FO), it follows that the integrals x x r'® f' (dt and at Jre Fo are either both bounded or both unbounded. Hence we conclude that the series MQ) Fn and 5 LL i Fo will either both converge or both diverge. In the case of divergence, when necessarily f(#) — + 00, we have =37 Le convergent when «> 1. In fact, here J Cg 1 1 gy —1 [f° e—1. [FOO whence the validity of the statement can be directly inferred. — These theorems are closely connected with the theorem of Abel-Dini. 2. A test of practically the same scope, and independent of the integral calculus in its wording, is Ermalkofi’s test®S. If f(x) is related to a given series 2 a, of positive, monotonely diminishing terms, in the manner described in the integral lest, and also satisfies the conditions there laid down, then 2 converges ete PC] | Za, =f) diverges | f(@) £5 1 for every sufficiently large x. Proof. If we suppose the first of these inequalities satisfied for x > x,, we have for these 2's Troae=fetrear x,, less than a certain fixed number. The series 2'g, must there- fore converge, by the integral test. If, on the other hand, we assume the second inequality satisfied for x > x,, we have, for these #’s, J rar= fo fe?) jar J rl) etl A comparison of the first and third integrals shows further that el Jroa=T roan. On the right hand side of this inequality, we have a fixed quantity y > 0, and the inequality expresses the fact that for every = (> x,) we can assign k, so that (with the same meaning for J as in 176) n+kn Jour lm ris 20. By 46 and 50, the numbers J, cannot be bounded and 2g, therefore cannot converge ??, Remarks. 1. Evmakoff’s test bears a certain resemblance to Cauchy’s condensation test. It contains, in particular, like the latter, the complete logarithmic com- parison scale, to which we have thus a fourth mode of approach. In fact, the behaviour of the series 1 nlogn... log, , n (log, n)® is determined by that of the ratio log, ,2-(log, z)* (log, 4 ®)* 27 It is not difficult to carry out the proof without introducing integrals, but it makes it rather more clumsy. 298 Chapter IX. Series of positive terms. As this ratio tends to zero, when «> 1, but — +00, when « <1, Ermakoff's test therefore provides the known conditions for convergence and divergence of these series, as asserted 28, 2. We may of course make use of other functions instead of ¢*. If ¢ (z) is any monotone increasing positive function, everywhere differentiable, for which ¢@ (x) > x always, the series Xa, will converge or diverge according as we have = ? @)f(p@) { Zo] f@) 21 for all sufficiently large a's. With Ermakoff's test and Cauchy's integral test, we have command over the most important tests for our present series. § 41. General remarks on the theory of the convergence and divergence of series of positive terms. Practically the whole of the 19th century was required to estab- lish the convergence tests set forth in the preceding sections and to elucidate their meaning. It was not till the end of that century, and in particular by Pringsheim’s investigations, that the fundamental questions were brought to a satisfactory conclusion. By these researches, which covered an extremely extensive field, a series of questions were also solved, which were only timidly approached before his time, although now they appear to us so simple and transparent that it seems almost inconceivable that they should have ever presented any difficulty ??, still more so, that they should have been answered in a com- pletely erroneous manner. How great a distance had to be traversed before this point could be reached is clear if we reflect that Euler never troubled himself at all about questions of convergence; when a series occurred, he would attribute to it, without any hesitation, the value of the expression which gave rise to the series®. Lagrange in 17703! was still of the opinion that a series represents a definite value, provided only that its terms decrease to 032. To refute the latter 28 This also holds for p= 0, if we interpret log_, z to mean e?. 29 As a curiosity, we may mention that, as late as 1885 and 1889, several memoirs were published with the object of demonstrating the existence of con- vergent series 3X ¢, for which “nt1 4id not tend to a limit! (Cf. 189, 3.) n 80 Thus in all seriousness he deduced from = =1+ax+a*+..., that Le1-tr1=14-.. and erie, Cf. the first few paragraphs of § 59. 81 V, (Euvres, Vol. 3, p. 61. 32 In this, however, some traces of a sense for convergence may be seen. § 41. General remarks on series of positive terms. 299 assumption expressly by referring to the fact (at that time already well known) of the divergence of I, appears to us at present superfluous, and many other presumptions and attempts at proof cur- rent in previous times are in the same case. Their interest is there- fore for the most part historical. A few of the questions raised, how- ever, whether answered in the affirmative or negative, remain of sufficient interest for us to give a rapid account of them. A con- siderable proportion of these are indeed of a type to which anyone who occupies himself much with series is naturally led. The source of all the questions which we propose to discuss resides in the inadequacy of the criteria. Those which are necessary and sufficient for convergence (the two main criteria 70 and 81) are of so general a nature, that in particular cases the convergence can only rarely be ascertained by their means. All our remaining tests (comparison tests or transformations of comparison tests) were suffi- ctent criteria only, and they only enabled us to recognise as conver gent series which converge at least as rapidly as the comparison series employed. The question at once arises: 1. Does a series exist which converges less rapidly than any other? 178. This question is already answered, in the negative, by the theorem 175, 4. In fact, when X'¢c, converges, so does X'¢/ = Y To though, FL obviously, less rapidly than Jc , as c :c,/ =t exil, 1 The question is answered almost more simply by J. Hadamard ®®, who takes the series X'¢/ = Src, = V7). Since ¢, =v, , — 7, the ratio ¢,:¢/=V7,., + Vr, — 0. The accented series conver- ges less rapidly than the unaccented series. The next question is equally easy to solve: 2. Does a series exist which diverges less rapidly than any other? Here again, the theorem of Abel-Dini 178 shows us that when Xd, diverges, so does 2'd,’ = 5 and hence the answer has to be in the negative. In fact as d,:d, = D,— + co, the theorem provides, for each given divergent series, another whose divergence is not so rapid. These circumstances, together with our preliminary remarks, show that 3. No comparison test can be effective with all series. Closely connected with this, we have the following question, raised. and also answered, by Abel3t: 8 Acta mathematica, Vol. 18, p. 319. 1894. 8 J. f. d. reine u. angew. Math., Vol. 3, p. 80. 1828. 300 Chapter IX. Series of positive terms. 4. Can we find positive numbers p,, such that, simultaneously, a) p,a,—0 oe bY pa Za>0 divergence of every possible series of positive terms? It again follows from the theorem of Abel-Dini that this is not | are sufficient conditions for the case. In fact, if we put a, = 3 o > 0, the series 2'a, necessar- n . . a ily diverges, and hence so does Za =X"7, where s, =a, +-+--+-a,. But, for the latter, p a,’ ==-0. n The object of the comparison tests was, to some extent, the con- struction of the widest possible conditions sufficient for the determination of the convergence or divergence of a series. Conversely, it might be required to construct the narrowest possible conditions necessary for the convergence or divergence of a series. The only information we have so far gathered on this subject is that ¢ — 0 is necessary for convergence. It will at once occur to us to ask: 5. Must the terms a, of a convergent series tend to zero with any particular rapidity? It was shown by Pringsheim?®® that this is not the case. However slowly the numbers p, may tend to 4-oco, we can invariably construct convergent serics 2'¢, for which limp, c, = -+- oo. - Indeed every convergent series X'¢,’, by a suitable rearrangement, will produce a series J¢_ to support this statement®®. Proof. We assume given the numbers p_, increasing to oo, and the convergent series 2c’. Let us choose the indices #n,, n,, ..., ns... odd and such that , I. < “3v=1 (v Pn, v and let us write C, = ¢3,—1, filling in the remaining c's with the terms =1,2,..) ¢,’s c/s... in their original order. The series 2'¢, is obviously a re- arrangement of X¢'. But Pn >” whenever # becomes equal to one of the indices #,. Accordingly, as asserted, lim p nn — + oo. The underlying fact in this connection is simply that the behaviour or a sequence of the form (p,¢,) bears no essential relation to that of | 35 Math. Annalen, Vol. 35, p. 344. 1890. : 38 Cf. Theorem 82,3, which takes into account a sort of decrease on the average of the terms a,. § 41. General remarks on series of positive terms. 301 the series 2c, — i. e. with the sequence of partial sums of this series, — since the latter, though not the former, may be funda- mentally altered by a rearrangement of its terms. 6. Similarly, no condition of the form limp d, > 0 is necessary for the divergence of 2'd,, however rapidly the positive numbers p_ may increase to -+o0o??. On the contrary, cvery divergent series 2d, provided its terms tend to 0, becomes, on being suitably rearranged, a series Xd_ (still divergent, of course) for which imp, d,=0. — The proof is easily deduced on the same lines as the preceding. The following question goes somewhat further: 7. Does a scale of comparison tests exist which is sufficient for all cases? More precisely: Given a number of convergent series Se, T.9...,5c9,... each of which converges less rapidly than the preceding, with e. g. ck+1) Sn + oo, for fixed k. 7; {The logarithmic scale affords an example of such series.) Is it pos- sible to construct a series converging less rapidly than any of the given series? The answer is in the affirmative®%. The actual construction of such a series is indeed not difficult. With a suitable choice of the indices %,, #,, +..> #;,...y the series =m (1) (1) (2) (2) (3) C,.=Ea + c2 + ta tat en on 1 eee : (3) (4) ttn, hpi is itself of the kind required. We need only choose these indices so large that if we denote by 72 the remainder, after the nth term, of the series 2 oY, for every n > n,, we have 7,® < 2 with 0.2 > 2, 0 1 ” ws Ly is Ny => Ny, no» 2,3 << og 7 6,2 > 2 £2 1 ” "un > Voy 0 kn <7 9% Eat >2 0, . . . . . . . . ° . . . . . . . . . . . . . . The series 3c, is certainly convergent, for each successive portion of it belonging to one of the series X¢,® is certainly less than the 37 Pringsheim, loc. cit. p. 357. 8 For the logarithmic scale, this was shewn by P. du Bois-Reymond (J. f. i. reine u. angew. Math., Vol. 76, p. 88. 1873). The above extended solution is due to J. Hadamard (Acta math, Vol. 18, p. 825. 1894). 11 302 Chapter IX. Series of positive terms. remainder of this series, starting with the. same initial term, i. e. < = (k=2, 3,...). On the other hand, for every fixed k, er in fact for n > n_(q > k) we have obviously = > 227% This proves po all that was required. — In particular, Hos are series converging more slowly than all the series of our logarithmic scale3?. 8. We may show, quite as simply, that, given a number of di- vergent series > 49, k=1, 2, ..., each diverging less rapidly than the preceding, with, specifically, d,, ins ==d, ® _, 0, say, there are always divergent series Xd, diverging Joss rapidly than every ome of the - series 2d, = All the above remarks bring us near to the question whether and to what extent the terms of convergent series are fundamentally distin- guishable from those of divergent series. In consequence of 7. we shall no longer be surprised at the following observation of Stieltjes: 9. Denoting by (e,, &,,...) an arbitrary monotone descending se- quence with limit 0, a convergent series Sc, and a divergent series 2'd, can always be specified, such that c, =c¢,d,. — In fact, if e,—0 monotonely, p_ i a + co monotonely. The series Prt —0) + HP — pn whose partial sums are the numbers p , is therefore divergent. By the theorem of Abel-Dini, the series : is also divergent. But the series Sc, = J¢ d =3(5- ] ) is 1, convergent by 131. — The following remark is only a re-statement in other words of the above: 10. However slowly pn—> +00, there is a convergent series Zc, and a divergent series Xd, for which d, =p, c,. In this respect, the two remarks due to Pringsheim, given in b. and 6., may be formulated even more forcibly as follows: 8 The missing initial terms of these series may be assumed to be each replaced by unity. § 41. General remarks on series of positive terms. 303 11. However rapidly 2'c, may converge, there are always divergent series, — indeed divergent series with monotonely diminishing terms of limit 0, — for which ood lim -* = 0. (x Thus 2'd, must have an infinite number of terms essentially smaller than the corresponding terms of X'¢ . Conversely: However rapidly 2d, may diverge, provided only d,— 0, there . . FC are always convergent series 2c, for which lim —= = -1 00. We have only to prove the former statement. Here a series Jd, of the form z 1 1 1 2 %=a+ 4 Ferg, + 5 tm Tun yon f= 1 1 i 1 +5% Tmt trom tyr is of the required kind, if the increasing sequence of indices n,, n,, ... be chosen suitably and the successive groups of equal terms contain respectively n,, (n, — n,), (n; — m,),... terms. In fact, in order that this series may diverge, it is sufficient to choose the number of terms in each group so large that their sum > 1, and in order that the se quence of terms in the series be monotone, it is sufficient to choose Mp > My, so large that ¢, 47 r=12,..) and write Cc = == C ! 1 = == TT ’ ny Pn, 1 mt =Cptac=1'r==0C; = Er = nay Pn, ¢ C 1 nn +15 wie Seite wae = ny = —» »—1 nV Pur The groups of terms here indicated contribute successively less than dos 1 ; 3 : >» ——3+.23—;... to the sum of the series 2c, -so that this series 2’ g u will converge. On the other hand, for each n =n, we have npc, = Vb.» so that, as was required, limun-p,-¢, =} cc. 13. These remarks may easily be multiplied and extended in all possible directions. They make it clear that it is quite useless to attempt to introduce anything of the nature of a boundary between convergent and divergent serics, as was suggested by P. du Bois- Reymond. The notion involved is of course vague at the outset. But in whatever manner we may choose to render it precise, it will never correspond to the actual circumstances. We may illustrate this on the following lines, which obviously suggest themselves *. a) As long as the terms of the series 2c, and X'd, are subjected to no restriction (excepting that of being > 0), the ratio - is capable n of assuming all possible values, as besides the inevitable relation lim —* = 0 we may also have im -* = + co, — even if it be required further that 4, — 0. The polygonal graphs ] by which the two sequences (c¢,) and (d,) may be represented, in accordance with 7, 6, can therefore intersect at an indefinite number 41 A detailed and careful discussion of all the questions belonging to the subject will be found in Pringsheim’s work mentioned on p. 2, and also in his writings in the Math. Ann. Vol. 35 and in the Miinch. Ber. Vol. 26 (1896) and 27 (1897), to which we have repeatedly referred. b § 42. Systematization of the general theory of convergence. 305 of points (which may grow more and more numerous, to an arbitrary extent). b) By our remark 11, this remains true when the two sequences (¢c,) and (d,) are both monotone, in which case the graphs above re- ferred to are both monotone descending polygonal lines. It is therefore certainly not possible to draw a line stretching to the right, with the property that every sequence of type (c,) has a graph, no part of which lies above the line in question, and every sequence of type (d,) a graph, no part of which lies below this line, — even if the two graphs are monotone and are considered only from some point situated at a suffi- ciently great distance to the right. These negative statements are not essentially altered if we assume the two sequences (c,) and (d,) not merely simply monotone as above, but fully monotone in the sense of p. 263*. §42. Systematization of the general theory of convergence. The element of chance inherent in the theory of convergence as developed so far gave rise to various attempts to systematize the criteria from more general points of view. The first extensive attempts of this kind were made by P. du Bois-Reymond*?, but were by no means brought to a conclusion by him. A. Pringsheim?*® has been the first to accomplish this, in a manner satisfactory both from a theoretical and a practical standpoint. We propose to give a short account of the leading features of the developments due to him‘. All the criteria set forth in these chapters have been comparison tests, and their common source is to be found in the two comparison tests of the first and second kinds, 157 and 158. The former, namely @) nSCu 3: C Op 2 1D, is undoubtedly the simplest and most natural test imaginable; not so that of the second kind, given originally in the form 1 And! oc Cut An41~ dni . . {0 Un 5 Cn 2, an 2 dn : 2 * H. Hahn, Uber Reihen mit monoton abnehmenden Gliedern, Monatsheft f. Math. u. Physik, Vol. 33, pp. 121—134, 1923. _ “J. fd. reine u. angew. Math. Vol. 76, p. 61. 1873. 43 Math. Ann. Vol. 85, pp. 297—3894. 1890. : 4 We have all the more reason for dispensing with details in this connexion, seeing Pringsheim’s researches have been developed by the author himself in a very complete, detailed, and readily accessible form, 306 Chapter IX. Series of positive terms. In considering the ratio of two successive terms of a series we are already going beyond what is directly provided by the series itself. We might therefore in the first instance endeavour to construct further types of tests by means of other combinations of two or more terms of the series. This procedure has, however, not yielded any criterion of interest in the study of general types of series. If we restrict our consideration to the ratio of two terms, it is still possible to assign a number of other forms to the criterion of the second kind; e. g. the inequalities may be multiplied by the positive factors a, or ¢, without altering their significance. We shall return to this point later. Except for these relatively unimportant transformations, however, we must regard (I) and (II) as the fundamental forms of all criteria of convergence and divergence*®. All conceivable special com- parison tests will be obtained by introducing in (I) and (II) all conceiv- able convergent and divergent series, and, if necessary, carrying out transformations of the kind just indicated. The task of systematizing the general theory of convergence will accordingly involve above all that of providing a general survey of all conceivable convergent and divergent series. This problem of course cannot be solved in a literal sense, since the behaviour of every series would be determined thereby. We can only endeavour to reduce it to factors in themselves easier to survey and therefore not appearing so urgently to require further treatment. Pringsheim shows — and this is essentially the starting point of his investigations — that a systematization of the general theory of convergence can be fully carried out when we assume as given the totality of all monotone sequences of (positive) numbers increasing to —- co. Such a sequence will be denoted by (p,); thus 00. With a few restrictions of little importance, all divergent and convergent series are also expressible in one of these new forms. 2. Since the only condition to be satisfied by the numbers p , in the typical forms of divergent and convergent series which we are 46 Unless the terms are all 0 from some stage on. 47 The proofs of these two statements are so easy that we need not go into them further. 308 Chapter IX. Series of positive terms = considering, is that they are to increase monotonely to -- co, we may of course write logp,,log,p,,... or generally F(p,) instead of Das where F(x) denotes any function defined for > 0 and increasing monotonely (in the strict sense) to + co with x. This again leads to criteria which, though not essentially new, are formally so when the p,’s are specially chosen. Itis easy to verify that the first named types of series diverge or converge more and more slowly, as p — +00 more and more slowly; by replacing p, sucessively e. g. by logs , log, p,, --., we therefore obtain a means of constructing scales of criteria *®. The case p =n naturally calls for consideration on account of its peculiar simplicity; the development of the ideas indicated above for this particular case forms the main contents of §§ 37 and 38. 3. A further advantage of this method is due to the fact that one and the same sequence (p,) will serve to represent both a di- vergent and a convergent series. The criteria therefore naturally occur in pairs. E. g. every comparison test of the first kind may be deduced from the pair of fests: = : e = Pa Ph-a Cn Dp 1 > n n— : 3 Pp—1 and similarly for other typical forms of series. 4. The right hand sides can be combined to form a single dis- junctive criterion, if we introduce a modification, arbitrary in character in so far as it is not necessarily suggested by the general trend of ideas, but otherwise of a simple nature. We see at once, for instance, that the series 2, Pu 2p lpr Sr and nn converge when ¢ > 1 and diverge when « <1. The pair of criteria set up in 3. may accordingly be replaced by the following disjunctive criterion: 19,7. ; a> 1 : Cir a, = | ——— L with > 2 py, a = 1 : dD 48 The usual passage from p, direct to log p,, log, p,, ..., is again quite an arbitrary step, of course. Theorems 77 and 175, 2 render the step natural, however. Between e.g. p, and logp,, we could easily introduce inter- mediary stages, for instance eV!°Pn which increases less rapidly than Pi; — in fact less rapidly than any fixed positive power of p,, however small its exponent, — yet more rapidly than every fixed positive power of log p,, however large its exponent. 2 % - § 42. Systematization of the general theory of convergence. 309 and, in all essentials??, also by: YP, — Un... ; a> e a,! =! —/————"- with > alr a1 : 9 It is remarkable that in the criteria of convergence arising through these transformations, the assumption p — + co is no longer necessary at all. It is sufficient that (p_) should be monotone. In fact, i (p,,) is boun- =P ded, the convergence of 2 (p, —p,._,), and hence that of slat 2 ond Sule for arbitrary « > 0, follows from that of (p,), as (p, i n > and (¢~?») are also bounded sequences. These convergence tests * thus possess a special degree of generality, similar to that of Kummer's®! cri- terion of the second kind, mentioned below in 7. 5. From this disjunctive criterion — as indeed in general from any criterion — others may again be deduced by various transformations, though the criteria so obtained can be new only in form. For these transformations we can of course lay down no general rule; new ways may always be found by skill and intuition. This is the reason for the great number of criteria which ultimately remain outside the scope of any given systematization. It is obvious that every inequality may be multiplied by arbitrary positive factors without altering its meaning; similarly we may form the same function F(x) of either member, provided F(x) be monotone increasing (in the stricter sense), — in particular we may take log- arithms, roots, etc. of either side. E. g. the last disjunctive criterion may therefore be put into the form 10g (pn — pn—1)—loga, [=> 0 e Pn =0 : 9D V/ a, cg] : ce Cr Cpl =>1 : a, We see at a glance that by this means we obtain a general frame- work for the criteria of the preceding sections which were set up by assuming p, =n or = log #n. or % The equivalence is not complete, i. e. with the same sequence (p,) as basis, the new criterion is not so effective as the old one; in fact, the divergence of >t a Fad. 2 , for instance, may be inferred from the old criterion, but not from hh new one, 5 Pringsheim: Math, Ann., Vol. 35, p. 342. 1890. 51 Journ. f. d. reine u. angew. Math,, Vol, 13, p. 78. 1835. : 11+ 310 Chapter IX. Series of positive terms. 6. Substantially the same remarks remain valid, when we sub- Pn — Pn n° Pn—1 terion of the second kind (II), or perform any of the other typical substitutions for ¢, and d,, there. In this way we obtain the most general form of the criteria of the second kind. stitute for ¢, and p, — p,_, for d, in the fundamental cri- 7. We may observe (cf. Rem. 4.) that here again, after carrying out a simple transformation, we may so frame the convergence test that it combines with the divergence test to form a single disjunctive criterion. The convergence test requires in the first instance that, for every sufficiently large =, pga Ruption nnd fay ol De Cn yy = Cn 8p Cpi1— 2 nec a , the former inequality reduces to n° Fn—1 If here we replace ¢, by Pri1—Pn Pn Pr Ap i1 Prni1 Pa Pn—Pn—1 an as p, cancels out, the typical terms of a divergent series automatically appear, so that the convergence test reduces to dps (-— 1) — 2 > 0 : e or dnt Ap 41 Sons St : dy a, — a, 41 C. Finally, if we take into account the fact that 2p d, (0 > 0) diverges with Xd , the criterion takes the form: 1 — Ap +4 1 dy an dy +1 20>0 > 1 e. Now the original criterion is certainly satisfied by the assumption It thus appears that in this form — slightly less general than the original form — of the convergence test, it is absolutely indifferent whether a convergent series or a divergent series is introduced as comparison series. Hence, still more generally, the ¢,’s and d's in the above forms of the criterion may be replaced by any (positive) numbers bd, ; thus we may write: b.— Put) Dri = 0 >0 : ec. Exercises on Chapter IX. 311 This extremely general criterion is due to E. Kummer ®? On the other hand, 1 41 1 Su : e 4, Ted 5 <0 9 represents a disjunctive criterion of the second kind which immediately follows, as the part relative to divergence is merely a slight trans- formation of (II). All further details will be found in the papers and treatise by A. Pringsheim. The sequences of ideas sketched above can of course lead only to criteria having the nature of comparison tests of the first or second kinds, though all criteria of this character may be developed thereby. The integral test 176 and Ermakoff's test 177d of course could not occur in the considerations of this section, as they do not possess the character in question. Exercises on Chapter IX. 133. Prove in the case of each of the following series that the given indications of convergence or divergence are correct: paint gs [lon a OE 9 I (10g 2D) : es 9S mre o FEEPletynbotd go [PZe20 i a 52 It was given by Kummer as early as 1835 (Journ. f. d. reine u. angew. Math., Vol. 13, p. 172) though with a restrictive condition which was first re- cognized as superfluous by U. Din: in 1867. Later it was rediscovered several times and gave rise, as late as 1888, to violent contentions on questions of priority. O. Stolz (Vorlesungen iiber allgem. Arithmetik, Vol. 1, p. 259) was the first to give the following extremely simple proof, by means of which the criterion was first rendered fully intelligible: Direct proof: The criterion is that from some stage on a,b, — Ap i1°bp41 = oa,. It follows in particular that the products a,b, diminish monotonely and therefore tend to a definite limit y >> 0. By 131, mle, Op — 8p 1170p 1g) 18 e thus a convergent series of positive terms. And as its terms are not less than ‘the corresponding terms of Ya, , this series is also convergent. 181. 312 Chapter X. Series of arbitrary terms. 134. For every fixed p, the expression % 1 === x! | vlogwv...log,» ogy ean | has a definite limit C, when n— 400, if the summation commences with the first integer for which log, n > 1. 135. For every fixed p in 0 << p <1, the expression 2 i] 2] 2 Ra Q has a definite limit 7 when # — 4+ 00. 136. If x, —&, it follows that x st en 1 pn+q Pp nt1)+q pp'ntq ’ ne 0. Dy 24 Snide os Pain) FR where p, p’, and g denote given natural numbers, 137. If 2d, is divergent, with d, — 0, and if the D,’s are its partial sums we have : % 1 dy Dy ~ 5 Dn. 3 y= 138. If Ya, has monotonely diminishing terms, it is certainly divergent when p-ayy —a, = 0 for a fixed p. and every sufficiently large =. 139. If 0 < 0. 140. Give a direct proof, without the use of Ermakoff's test and without the help of the integral calculus, of the criterion pe fp (Cl 28 lim ~>2 : 9 for series of monotonely diminishing terms. an 141. If the convergence of a series Xa, follows from one of the criteria of the logarithmic scale 164, II, then, as n — + 00, [nlognlog,n...logyn]-a,—0 and diminishes monotonely from a certain stage on, whatever the value of the positive integer 2 may be. Chapter X. Series of arbitrary terms. § 43. Tests of convergence for series of arbitrary terms With series of positive terms, the study of convergence and divergence was capable of systematization to some extent; in th i case of series of arbitrary terms, all attempts of this kind have to be abandoned. The reason lies not so much in insufficient de. § 43. Tests of convergence for series of arbitrary terms. S15 velopment of the theory, as in the essence of the matter itself. A series of arbitrary terms may converge, without converging abso- lutely!. Indeed this is practically the only case which will interest us here, as the question of absolute convergence reduces, by 8%, to the study of a series of positive terms. We therefore need only consider the case in which either the series is actually not absolutely conver- gent or its absolute convergence cannot be demonstrated by any of the previously acquired means. If a series is conditionally conver- gent, however, this convergence is dependent on the mode of succession of the terms as well as on their individual values; any comparison test which we might set up would therefore have to concern the series as a whole, and not merely its terms individually, as before. This ultimately means that each series has to be examined by itself and we cannot obtain a general method of approach valid for them all Accordingly we have to be content to establish criteria with a more restricted field of validity. The chief instrument for the purpose is the formula known as Cdbel’s partial summation®. If ay, a,,... and by, b,, ... denote 182. arbitrary numbers, and we write do +01 va, =4, (n> 0) then for every n> 0 and every k > 1, n+4-k n+k 2 b,= 2 A D,— Vi )— An Oni1 + Anite On 041. Proof. We have a, b, = (4, TE 4,0 b, = 4.0 = Dyin) or Aq b, = 4,0,513 by summation from » =n +1 to vy =n -}- k, the statement at once follows?®. OSupplements. 1. The formula continues to hold when n= — 1, 183. if we put 4_,=0. ! The case in which the series may be transformed into one with posi- tive terms only, by means of a “finite number of alterations” (v. 82,4) or by a change of sign of all its terms, of course requires no special treatment, 2 Journ. f. d. reine u. angew. Math. Vol. 1, p. 314. 1826. 8 It is sometimes more convenient to write the formula in the form n+k nt+k—-1 ol ay by = 2 Obi) = Aba t Arnley n v=n+1 y= 184. 314 Chapter X. Series of arbitrary terms. 2. If c¢ denotes an arbitrary constant, and A] =A, +c, we have also: n+k n+k J ab= 3 A’ 0, —b41)—A.0np1+ A345 Dugrsr v=n+41 v=n+1 fora =A — 4, j=4 — Al, Accordingly, in Abel's partial summation we “may” increase or diminish all the 4's by any constant amount. This is equivalent to altering a. Abel's partial summation enables us to deduce a number of tests of convergence for series of the form 2g b, almost immediately * In the first place, it provides the following general OTheorem. The series 2a, b, certainly converges, if 1) the series 2 A,(b,—b,,,) converges, and 2). ImA 8, exists. P>+ x Proof. hors partial Samana gives for n = — 1: Tap = 34 (0, — b, 11) + 43 bya for every k > 0; making : — -- 00, the statement follows, in view of the two hypotheses. — The relation just written down shows further that S=¢g-]-1 where cab. =a, TAD Db )=5 lim4,b,,,=1. In particular, s=1¢ if, and only il, {=0. The theorem does not solve the question as to the convergence of the series 2a b,, since it merely reduces it to two new questions; but these are in many cases simpler to treat. The result is in any case a far- reaching one, and it enables us immediately to deduce the following more special criteria, which are comparatively easy to apply. O1. Abel's test’. a,b, is convergent if Za, converges and (b,) ts monotone and bounded®. 4 We can of course reduce any series to this form, as any number can be expressed as the product of two other numbers. Success in applying the above theorem will depend on the skill with which the terms are so split up. 5 loc. cit. — Abel's test provides a sufficient condition to be satisfied by (b,), in oder that the convergence of Xa, may involve that of Xa, b,. J. Hadamard (Acta math., Vol. 27, p. 177. 1903) gives necessary and sufficient conditions; cf. E. B. Elliot (Quarterly Journ., Vol. 37, p. 222. 1906). 8 In other words: A convergent series “may” be multiplied, term by term, by factors forming a bounded and monotone sequence. — Theorem 184 and the criteria deduced from it all deal with the question: By what factors may the terms of a convergent series be multiplied, and by what factors must the terms of a divergent series be multiplied, so that the resulting series may be convergent? § 43. Tests of convergence for series of arbitrary terms. 915 Proof. By hypothesis (4,) and (b,), (v. 46), and hence also (4,0, ry are convergent. On the other hand, by 181, the series >(b,—0b,,,) is convergent, and indeed absolutely convergent, as its terms all have the same sign, in consequence of the monotony of (b,,). It follows, by 87, 2, that the series 24, (b, — b,,,) is also convergent, since a convergent sequence is certainly bounded. The two conditions of theorem 184 are accordingly fulfilled and 2g, 6b, is convergent. - 02. Dirichlet’s test’. a,b, is convergent if 2 a, has bounded partial sums and (b,) is a monotone null sequence. Proof. By the same reasoning as above, 24, (b, — b,,,) is con- vergent. Further, as (4,) is bounded, (4,0, ,,) is a null sequence if (b,) is, i.e. it is certainly convergent. The two conditions of 184 are again fulfilled. 03. Tests of du Bois-Reymond® and Dedekind?. a) a,b, is convergent if 2(b,—b,,,) converges absolutely and 2a, converges, at least conditionally. Proof. By 87,2, 2A4,-(b,—b,,,) also converges, as (4,) is certainly bounded. Since further Go OYE 6, 0) Lo ib, = Bel b, tends to a limit when #-— -}-0c0, so does b, itself; lim 4, exists by hypothesis, and the existence of lim 4, b,,, follows. b) 2a,b, is convergent if 2(b, —b,,,) converges absolutely and 2a, has bounded partial sums, provided b,— 0. Proof. 2A4,(b,—b,,,) is again convergent and 4b , ,—O- n or Examples and Applications. 185 1. The convergence of Xa, involves, by Abels test, that of Sri, n a n+ 1 = 1\» an Zon > n %n Yn, (1+) ‘a, etc. 2. 2 (— 1)* has bounded partial sums. Hence if (b,) is a monotone null sequence, (=n ? Vorlesungen iiber Zahlentheorie, 1® edition, Braunschweig 1863, § 101. 8 Antrittsprogramm d. Univ. Freiburg, 1871. — The designation above adopted for the three tests is rather a conventional one, as all three are sub- stantially due to Abel. For the history of these criteria cf. A. Pringsheim, Math, Ann, Vol. 25, p. 423. 1885. ® § 143 of the work referred to in footnote 7. ”n — 10 For Vn diminishes monotonely from » = 3, but remains > 1. 316 Chapter X. Series of arbitrary terms. converges by Divichlet's test. This is a fresh proof of Leibniz's criterion for series with alternately positive and negative terms (82,5). 5 8. Given positive integers &,, &,, k,, ... such that > (— 1)%n has bounded partial sums — for this the excess of the number of even integers over that of odd integers among the # first exponents %,, kg, «.., k, has to remain bounded as n— + 00 — the series S(=D, converges, if (b,) denotes any null sequence. 4. If Xa, is convergent, the power series: J a,x” Is convergent for 0 cos# xz have bounded partial sums, the first for every (fixed) real x and the second for every (fixed) real x not a multiple of 2x. This follows from the following elementary but important formula, valid for every x + 2k a: sin ny sin (« + (n+1) 7 3 ; : 2 sin (+a) +sin(¢ +22) +--+ +sin(e4+n2)=——————————=, HU sin & The proof of the formula is given in 201. For «=0, we get sin n - sin(n +1) 2 5 singtsin2edtsimmg=— > 2 @F2%7) sin — “ 2 and for @=g, sin n 3 cos m+) coszx+cos2x+..-tcosng=——————, (x+2Fka). sin 5 From this the boundedness of the partial sums can be inferred at once. Thus if > (b, —b, +1) converges absolutely and b, — 0, we conclude from the criterion 3b that 2b, sinnx converges for every x, 2b, cosnax converges for every x += 2k x. In particular, this is the case when b, diminishes monotorely to. {22 6. If the b,’s are positive, and if we may write where § >> o and (8,) is bounded, then 3 (— 1)» b, converges if, and only if, «> 0. In fact, if « > 0, it follows from these hypotheses that ents <1 from some stage on, i. e. (b,) decreases monotonely, and the convergence of the series in question is therefore secured by 2. if we can show that b,— 0. The proof of this is similar to that of the parallel fact in 170, 1: If 0 < &’ << «, we have for every sufficiently large », say » = m, ’ by 41 o a r 11 For # = 2% =n, the sum has obviously the value # sin «, for all ns. 12 Malmstén, C. J.: Nova acta Upsaliensis (2), Vol. 12, p. 255. 1844. Ne § 43. Tests of convergence for series of arbitrary terms. S17 Writing down this inequality for v=m, m+1, ..., n—1 and multiplying together, we obtain n-1 o 52h, TT (1-2). Y=m From the divergence of the harmonic series, it follows as in 170, 1 that b,—0. In the case a <0, b, must for similar reasons increase monotonely from some stage on, so that X(— 1)”b, certainly cannot converge. Finally, when a= 0, we deduce in precisely the same way as on p. 289, that b, cannot tend to ( and the series therefore cannot converge. 2 a. i; Ta 7. If a series of the form as — such series are known as Dirichlet series; we shall investigate them in more detail later on (§ 58, A) — is con- vergent for a particular value of z, say x =z,, it also converges for every zz 2,, for ( ) is a monotone null sequence. This simple application of n ‘0 Abel's test, by reasoning quite similar to that employed for power series (93), ; a ; J leads to the theorem: Every series of the form > 2 possesses a definite abscissa of convergence A with the property that the sevies converges whenever x > ) and diverges whenever x > 0, and = 0 when a,, < 0, and similarly let Tn=—0, when a, <0, and = 0 when a, => 0.1% The two series 3p, and X¢, are series of positive terms, the first containing only the positive terms of 2a, and the second only the absolute values of the negative terms of Xa,, in either case with the places unchanged, while their other terms are all 0. Both these series are divergent. In fact, as every partial sum of Xa, is the difference of two suitable partial sums of 2p. and Xg,, it follows at once that if Xp, and Xg¢, were both convergent, so would 3 |a,| be (by 70), contrary to hypothes's; . and if the one were convergent, the other divergent, the partial sums of dy — 8 Thus p, = Sk AR phen ct. p. 139, footnote 16. 2 318 Chapter X. Series of arbitrary terms. 187. would tend to — 0 or + 00 (according as Xp, or 2g, is assumed convergent), which is again contrary to hypothesis. 3. By the preceding remark, a conditionally convergent series, or rather the sequence formed by its partial sums, is exhibited as the difference of two monotone increasing sequences of numbers tending to infinityl4. As regards the rapidity with which these increase, we may easily establish the following Theorem. The partial sums of 2p, and 2g, ave asymptotically equal. In fact, we have Pit 0:11 Pn iat ht ia, Gi+dat tn G+dat- tan’ since the numerator in the latter ratio remains bounded, while the denominator increases to -+ 00 with #, this ratio tends to 0, which proves the result. 4. The relative frequercy of positive and negative terms in a conditionally convergent series Xa, for which |a,| diminishes monotonely is subject to the following elegant theorem, due to E. Cesdro'®: The limit, if it exists, of the LE % 2 ; ratio —" of P,, the number of positive teyms, to Q,, the number of negative terms a,, n for » , and yu, > 0.27 Let us denote by 9, 4,;.:1 the tens IN" Ja mg, + a, =r: which are => 0, in the order in which they ocewr, and by ¢q,, 4,.--- the absolute values of those which are << 0, again in their proper order, thus slightly modifying the definition in 186, 2. The series 2p, and 2g, only differ from those in 186, 2 by the absence of a number of zero terms, and are accordingly both divergent, with posi- tive terms which tend to 0. We proceed to show that a series of the type Pa Dn Lr = baat Pe — uggs a ssomitly ob By, pq leas will satisfy all the requirements. Such a series is clearly a re- arrangement of the given series, and is indeed one which leaves un- altered the order of the positive terms relatively fo ome another and that of the negative terms relatively fo ome another. Let us. choose the indices mp, <1, < ies. By By << very In the above series, so that: 1) the partial sum whose last term is p,, has a value > Hy while that ending one term earlier is < u,; 2) the partial sum whose last term is — g;, has a value < x, while that ending one term earlier is i ny; 3) the partial sum whose last term is p,, has a value > u,, while that ending one term earlier is < u,; 16 Riemann, B.: Abh. d. Ges. d. Wiss. z. Gottingen, Vol. 13, p. 97. 1866—68. The statements b) and c) are obvious supplementary propositions. 17 This is clearly possible in any number of Ps In fact, if x =u with 1 a finite value §’, say, take x, wile and yu, =35 31, — taking u, even larger, if necessary. If x=pu=+ 00 (—00), take »,=n on wy: and pg, =r, +2. If finally, » 0. i 320 Chapter X. Series of arbitrary terms. 4) the partial sum whose last term is — g;, has a value < x,, while that ending one term earlier is > x,; . and so on. This can always be arranged; for by taking a sufficient number of positive terms, the partial sum may be made as large as we please, and by allowing a sufficient number of negative ones to follow, the partial sum may again be depressed below any assigned value. Let Ja ' denote the definite rearrangement of Za, so obtained; the partial sums of 3a have the prescribed upper and lower limits. In fact, if for brevity we denote by 1,, 7,,..., the partial sums whose last terms ave pp, , Py, .-- 80d by o,, 0,,..., those whose last terms are = gr, — Gp, ---, we have ja, =] < qr, and |%, —n. < Pm, Since p,— 0 and ¢,— 0, it follows that ¢,— x» and 7,— u, so that » and wu certainly represent values of accumulation of the partial sums of Xa . Now a partial sum s/ of Ja /, which is neither a g, nor a 7,, has necessarily a value between those of two successive partial sums of this special type; hence s,’ can have no value of accumulation outside the interval x...u, (or different from the common value of » and pu if these coincide). In other words, 4 and x» are themselves the upper and the lower limit of the partial sums, q.e.d. Various researches of an analogous nature were started in different direc- tions as a consequence of this theorem. M. Ohm?'® and O. Schlomilch!® investi- gated the effect of rearrangement on the special series 1 —1y Tiled, in particular the case in which p positive terms are followed by ¢ negative terms throughout (cf. Exercise 148). A. Pringsheim?®?) was the first, however, to aim at general results for the case in which the relative frequency of the posi- tive and negative terms in a conditionally convergent series is modified accord- ing to definite prescribed rules. E. Borel?! investigated the opposite problem, as to what rearrangements in a conditionally convergent series do not alter its sum. Later, W. Sierpiniski?® showed that if Xa, =s converges conditionally and s’ s by rearrang- ing only the negative terms. (The proof is not so simple.) § 45. Multiplication of conditionally convergent series. We showed in the preceding section, thus completing the con- siderations of 89,2, that the commutative law of addition no longer holds for series which converge only conditionally. We have also seen 18 Antrittsprogramm, Berlin, 1839. 18 Zeitschr. {. Math. u. Phys., Vol. 18, p. 520. 1873. 20 Math. Ann., Vol. 23, p.455. 1883. 2 Bulletin des sciences mathém. (2), Vol. 14, p. 97. 1890. 22 Bull. internat. Ac. Sciences Cracovie, p. 149. 1911. § 45. Multiplication of conditionally convergent series. 521 already (end of § 17), in an example due to Cauchy, that the dis- tributive law does not in general subsist, so that the product of two such series X'q¢_ and 2b, may no longer be formed according to the elementary rules. The question remained unsolved, however, whether the product series J¢, (withe¢,=a,b, 44,5, , +--+ a,b,) might not continue to converge under less stringent conditions for Xa = 4 and 2b = B, and to have the sum 4.B. In §17, it was required that both 2g, and Xb, should converge absolutely. In this connection, we have first the O Theorem of Mertens®®. If one at least — say the first — of 188. the two convergent series 2a, = A and 2b, = B converges absolutely, 2c, converges and = A-B. = Proof. We have only to show that, with increasing =, the partial sums : C.=0"1¢ zh 4s +e, = ay by + (#0, +2, 85) +++ + (4,0, + a,b, +--+ a,b) tend to 4-B as limit. If we denote by A, the partial sums of 2a, by B, those of 2b, we have C.=ayB, 0B irr 8, By or, if we put B,=B-8, = 4,8 + (ay: B, ote a, 8, = te + a, Bo): Since 4 -B—A-B, it only remains to show that when Jag is absolutely convergent and f, — 0, the expressions ©, =a, B+ a, 1 wish rele dy By form a null sequence. But this is an immediate consequence of 44, 9b; we have only to put #, =p, and y, = gq, there. Thus the theorem is proved. Finally, we shall answer the question whether the product series 2c,, if convergent, necessarily has the sum A-B. The answer is in the affirmative, as the following theorem shows: OTheorem of Adbel®:. If the three series 2a,, 2b, and 189. 2c, =2(ayb, +--+ a,b, are convergent, and A, B, and C are their sums, we have A-B = C. 1. Proof. The theorem follows immediately from Abel's limit theorem (100) and was first proved by Abel in this way. If we write Za," =f, (2), 2b," = f,(), Ze," = fy (v), 28 J. f. d. reine u. angew. Math., Vol. 79, p. 182. 1875. — An extension was given by T. J. Stieltjes (Nouv. Annales (3), Vol. 6, p.210. 1887). - 2 J, f. d. reine u. angew. Math., Vol. 1, p. 318. 1826. 322 Chapter X. Series of arbitrary terms. these three power series (cf. 185,4) certainly converge absolutely for 0< 2 <1, and for these values of x, the relation (a) f1(®)- fo (x) = f(x) holds. The assumed convergence of Xa, , 2b, and 2¢, implies, by Abel's limit theorem 100, that each of the three functions tends to a limit when 2 — + 1 from the left; and % (®)—>A4=2a, fo (®)—>B=2b,, f;(®)—C=2c,. Since the relation (a) holds for all the values of x concerned, it follows (by § 19, Theorem 1) that it must hold in the limit: A B==C. — We may also dispense with the use of functions and adopt the following 2. Proof due to Cesdro?®. It was shown above that C,=aB-19B_ ++ +{+aB; From this it follows that’ CoC, tr" C = AB 1 A,.8, 17+ 4.8, Dividing both sides of this equality by # 1 and letting #— + oo, we obtain C as limit on the left hand side (by 43,2) and 4-B as limit on the right (by 44,9a). Hence 4-B=C, q.e.d. In consequence of this interesting theorem, with which we shall again be concerned later on, any further elaboration of the question of multiplication of series has only to deal with the problem whether the series 3c, converges. Into these investigations we do not, however, propose to enter? Examples and Applications. 1. Tt follows from ado > n= i =1—-—+4+———+4..., by the pre- 2u-+1 3 5 7 ceding theorem, that we 30 (fart amet Fa) 6 = 1.2n+1) ' 8.2n—1) @2n+1)-1)° provided the series thus obtained converges. 25 Bull. des sciences math. (2), Vol. 14, p. 114. 1890. 26 Theorems of the kind in question have been proved by A. Pringsheim (Math. Ann., Vol. 21, p. 340. 1883), and in connection with the latter's work, by A. Voss (ibid. Vol. 24, p. 42. 1884) and F. Cajori (Bull. of the Americ. Math. Soc., Vol. 8, p.231. 1901-2 and Vol. 9, p. 188. 1902-3). — Cf. also § 66 of A. Prings- heim’s treatise, Vorlesungen liber Zahlen- und Funktionenlehre (Leipzig 1916), to which we have already referred more than once. G.H. Hardy (Proc. Lon- don Math. Soc. (2), vol. 6, p. 410, 1908) has proved a particularly elegant example of a related group of much more fundamental theorems. 4 § 45. Multiplication of conditionally convergent series. 323 Now 1 1 rl ; 1 @p+D) @nt1-2p 2m+1)\2p+1" Trl) so that the generic term of the new series has the value =Ln 1 n+ 1 (1 ty +: pt). tends monotonely to zevo, so does its arithmetic mean 1 1 palit and the new series therefore does converge by Leibniz's test 82,5. We thus have SL pL primletyel (a) PH I++ i py =1 2. In a precisely similar manner, we deduce (v. 120), by squaring the Since 2n+1 series log 2 = 1—tal-3.., yt 1) 1 1 (b) ST 5 (14g +--+) = Cog 2 8. The result ie in 1. provides a fresh mode of approach to the 00 2 equation Sis7 which has occupied us repeatedly before now (v. 136 and 156). To see this, we first prove the following Theorem. Let (ay, a,, ay, ...) be a monotone sequence of positive numbers, for which 2 a2 18 A Then the series i. 3 ire =} 2 Stig mt ml, Dees n=0 and > A 1? op=4, p=1 also converge, with (c) Sat=s2_24. n=0 Proof. Since Xa,? converges, a,— 0; accordingly the series 1 con- verges by Leibniz's test. As a, a, < a,” for every p > 1, and Ya,? converges, the series 2 are also convergent for p > 1. Further, as py ppy S App, WE have 0,,,<4,. The series 3 will accordingly ii if 8,0. Now given ¢ > 0, we can choose m so that 22 ated deen <5 for every suffi- ciently large p, we shall then have Op < y+ 8, Gyr +o + Gn Gp rm to <8 (@y+ a+ + + a) +5 0, we have (cf. 8lc, 3) | Tu — Sa |S, +60 S Ont Fins thus, as was asserted, T,, — S, — 0 and therefore X'a,?=52—24. 4. If, in 3., we now take a, = the hypotheses are obviously all 1 2#+1° fulfilled, and we have = 1 72 = SUTTER TI thd But in this case, we have, by 133,1, = 1 1 1 5% oar Re he 25 (1 7) p= 3 errr nl rire for every p=1, and Bones Sryeat aol DE (1h gosta) re @2n+ 1)? -5° — n-41 er 2 By the equality (a) proved in 1., the right hand side — By the method oo 2 used to deduce 137 from 136, the equality FS i=- follows at once. k=1 - The fresh proof thus obtained for this relation may be regarded as the most elementary of all known proofs, since it borrows nothing from the theory of functions except the inverse tangent series 121. The main idea of the proof goes back to Nicolaus Bernoulli®". Exercises on Chapter X. : 142. Determine the behaviour of the following series: [Va] [Vn] 20 el © 9 3 ky ny 3 ol 0) za (22), Q Zin, 27 Comment. Ac. Imp. scient. Petropolitanae, Vol. X, p. 19. 1738. pi g RIN A aa as dg Ny Exercises on Chapter X. 325 e) Z(=1)rsin-, f) Isnt 2, g) 2 sin(®?%), h) 2 sin (n! =z), =n» sin? nx 9) ZT 2 BE 1\sinnz y . 1) Sty tet) = m) 2 a, sin nx cos? nx. In the last series, (¢,) is amonotone null sequence, The series g) does not converge unless x — k x; the series h) converges for all rational values of x, also e.g. for g=2, =(2%+11e, 2k, =sinl, =cosl, and for 1.1 Lilo] wll Slvr piT roo and many other special values of x. Indicate values of x for which it cer- tainly diverges. x 1 1 143. 2b ttre ra) — EY z= — Fee for every x > 0. 144. If (na,) and Xn (a, —a,,) converge, the series a, also con- verges. 145. a) If Ya, and X'|b,—b, | both converge, or b), if 3a, has bounded partial sums, |b, —b,4,| converges and b,— 0, then for every integer p=1 the series a,b, is convergent. 146. The conditions of the test 184,83 are in a certain sense necessary, as well as sufficient, for the convergence of Xa, b,: If it be required that for a given (b,), 2 a,b, always converges with Xa,, the necessary and sufficient condition is that X'|b,—b,, | should converge. — Show also that it makes little difference in this connection whether we require that X'|b, —b,,, | con- verges or merely that (b,) is monotone. 147. If Xa, converges, and if p, increases monotonely to 4 00 in such a way that 2p,~?! is divergent, we have Tahini: ‘tpn [ 20, raz 7 <0. 148. Let a, tend to 0 monotonely, and assume that lim na, exists. If we write Y= 1)" a, =s, and now rearrange this series (cf. Ex. 51) so as to n=0 have alternately p positive and g negative terms: Gta, +t ap_g—a—a3—-—ay,_ + a,+..., the sum s’ of the new series satisfies the relation To 3 = s+ lim (n a,)-log Z. 149. A necessary and sufficient condition for the convergence of the product series 2e,=2 (yb, +a, b,_1+---+a,b) of two convergent series Xa,, 3b,, is that the numbers n = On = 2 ay But bast et buyia) : od should form a null sequence. 326 Chapter XI. Series of variable terms. 150. If (a,) and (b,) are monotone sequences with limit (0, the Cauchy's product series of 2 (—1)”a, and 3 (—1)"b, is convergent if, and only if, the numbers o, =a, (by+b,+---+0,) and z,=0b,(a,+a,+---+a,) also form a null sequence. x 151. The two series Solr 1ye and = a0, 50, may be n* n multiplied together by Cauchy’s rule if, and only if, « +> 1. 152. If (a,) and (b,) are monotone null sequences, Cauchy's product of the series 3 (— 1)" a, and 2 (— 1)" b, certainly converges if a,b, converges. A necessary and sufficient condition for the convergence of the product series is that 2 (a, 5) should converge for every o> 0. 153. If, for every sufficiently large #, we can write a, = 1%. (log n)® . (log, n)*: ... (dog, n)*r On = who. (log n)f - (log, nf... (logs ns S and if Xb, converges, we have, provided a, is not equal to b, for every u, (@g by +a, bashed) Sa So,). Chapter XL Series of variable terms (Sequences of functions). § 46. Uniform convergence. Thus far, we have almost exclusively taken into consideration series whose terms were given (constant) numbers. It was only in particularly simple cases that the value of the terms depended on the choice of a definite quantity, or variable. Such was the case e. g. when “we were considering the geometric series 3a” or the harmonic series 1 3 . ! > —; their behaviour was dependent on the choice of a or of «. A more n general example is that of the power series Xa 2", where the number x had to be given, before we could attack the problem of its con- vergence or divergence. This type of case will now be generalized in the following obvious way: we shall consider series whose terms depend in any manner on a variable x, i. e. are functions of this variable. We accordingly denote these terms by f, («) and consider series of the form 2'f, (x). A function of x, in the general case, is defined only for certain values of & (v. § 19, Def. 1); for our purposes, it will be sufficient to assume that the functions f, (x) are defined in one or more (open or closed) intervals. For the given series to have a meaning for any value } 1 § 46. Uniform convergence. 827 of x at all, we have to require that at least ome point ® belongs to the intervals of definition of all the functions f, (x). We shall, however, at once lay down the condition that there exists at least one interval, in which all the functions f, (x) are simultaneously defined. For every particular « in this interval, the terms of the series Xf, (x) are in any case all determinate numbers, and the question of its convergence can be raised. We shall now assume further that an interval J (possibly smaller than the former) exists, for every point of which the series Jf, (x) is found to converge. O Definition. An interval ] will be called an interval of conver-190. gence of the series Xf, (x) if, at every ome of its points (including one, both, or meither of iis endpoints), all the functions f,(x) are defined and the series converges. Examples and Illustrations. 1. For the geometric series Xz” the interval —1 < «<1 is an interval of convergence, and no other interval of convergence exists outside it. 2. A power series Xa, (x — x,)", — provided it converges at one point at least, other than a, — always possesses an interval of convergence of the form (0g — 7)... (xy+ 7), inclusive or exclusive of one or both endpoints. When 7 is properly chosen, no further interval of convergence exists outside that one. : : ; 1 g : : 3. The harmonic series 2 has as interval of convergence the semi- n axis ¢ > 1, with no further interval of convergence outside it. 4. As a series is no more than a symbolic expression for a certain se- quence of numbers, so the series Xf}, (¥) represents no more than a different symbolic form for a sequence of functions, namely that of its partial sums sSn@=fo@+f@) + +fa(®)- In principle, it is thevefore immaterial whethey the terms of the sevies ov its partial sums ave assigned, as each set determines the othev uniquely. Thus, in principle, it also does not matter whether we speak of infinite series of variable terms or of sequences of functions. We shall accordingly state our definitions and theorems only for the case of sevies and leave it to the student to formulate them for the case of sequences of fumctions®. 5. '1f (z®)" Sp (X) = ———— a ( ) 1 on (x3)? ’ ! For the case of complex numbers and functions, we have here to sub- stitute throughout the word region for the word interval and boundary points of the region for endpoints of the interval. With this modification, the sign o has the same significance in this chapter as previously. 2 Occasionally, however, the definitions and theorems will also be applied to sequences of functions. 328 Chapter XI. Series of variable terms. so that our series is 3 f= + SS )+( > = ) eens 1422 1-|1=z° 1:28 142% _ this converges for every real x. Clearly 8) 5, @)—>0, if [o/<1, b) 5, ()—>1, if [2] 1, and 1 c) Sn (@) => 5, if jal=1. 6. On the other hand, S$, (x) = (2 sin)” - = defines a series with an infinity of separate intervals of convergence; for lim s, (z) obviously exists if, and only if, = sinz < 1 i.e if =O oa 4 : Sx 17 Si o —_—< Br %. povay or if x lies in an interval deduced from these by a displacement through an integral multiple of 2x. The sum of the series = (0 throughout the interior of the interval and =1 at the included endpoint. sina sin3w 2 3 z cos2x cos 3x real x; the series cosz + = 3 x=2%5. : If a given series of the form Xf (x) is convergent in a deter minate interval J, there corresponds to every point of J a perfectly definite value of the sum of the series. This sum accordingly (§ 19, Def. 1) is itself a function of x, which is defined or represented by the series. When the latter function is the chief centre of interest, it is also said to be expanded in the series in question. In this sense, we write 7. The series sin x 4 +... converges, by 185, 5, for every +... converges for every real F@)=31,@. In the case of power series and of the functions they represent (v. Chapters V and VI), these ideas are already familiar to us. : The most important question to be solved, when a series of variable terms is given, will usually be whether, and to what extent, properties belonging to all the functions f, (z), i e. to the terms of the given series, are transferred to its sum. : ! Even the simple examples given above show that this need not ; be the case for any of the properties which are of particular interest in the case of functions. The geometric series shows that all the func- tions f, (x) may be bounded, without F(x) being so; the power series for sinz, * > 0, shows that every f, (x) may be monotone, without F(z) being so; example 5 shows that every f, (x) may be continuous, § 46. Uniform convergence. 329 without F(z) being so, and the same example illustrates the corres. ponding fact for differentiability. It is easy to construct an example showing that the property of integrability may also disappear. For instance, let =1 for every rational x expressible as a fraction with denominator Sp (2) (positive and) < =, =(0 for every other z. Then s, (x), for each », — and consequently f, (x), for each n, — is inte- grable over any bounded interval, as it has only a finite number of discon- tinuities in such an interval (cf. § 19, theorem 13). Also lim s, (x) = F(x) exists for every xz. In fact, if x is rational, say -Z (¢g>0, p and g prime to one another), we have, for every 29, 5,(#)=1 and hence F@=1. If, on the other hand, x is irrational, s, (x) =0 for every » and so F(x) =0. Thus 2 fu (x) =lim s, (x) defines the function F () This function is not integrable, for it is discontinuous for every z 2 =1 for a rational =m, =( for an irrational x. Even by these few examples, we are led to see that a quite new category of problems arises with the consideration of series of variable terms. We have to investigate under what supplementary con- ditions this or the other property of the terms f, (x) is transferred to the sum F(x). Itis clear from the examples cited that the mere fact of convergence does not secure this, — the cause must reside in the mode of convergence. A concept of the greatest importance in this respect is that known as uniform convergence of a series 2'f, (x) in one of its intervals of convergence or in part of such an interval. This idea is easy to explain, but its underlying nature is not so readily grasped. We shall therefore first illustrate the matter somewhat intuitively, before proceeding to the abstract formulation: D0 Let 3 f, (x) converge, and have for sum F(z), in an interval J, a » F (x)= lim [ lim (cos®n! mz 2)¥]. n>» k->w This curious example of a function, discontinuous everywhere, yet obtainable by a repeated passage to the limit from continuous functions, is due to ~ Dirichlet. 330 Chapter XI. Series of variable terms. as the limiting curve. The fact of the convergence of 2 (x) to F(a) in | z n=0 then appears to imply that for increasing #, the curves of approximation lie closer and closer to the limiting curve. This, however, is only a very imper- fect description of what actually occurs. In fact, the convergence in J implies - only, in the first instance, that at each individual point there is convergence; all we can say, to begin with, is therefore that when any definite abscissa x is singled out (and kept fixed) the corresponding ordinates of the curves of \ approximation approach, as » increases, the ordinate of the limiting curve for the same abscissa. There is no reason why the curve y =s, (x), as a whole, should lie closer and closer to the limiting curve. This statement sounds rather paradoxical, but an example will immediately make it clear. The series whose partial sums for n=1, 2, ... have the values ° nx W@) = Te certainly converges in the interval 1 <2 <2. In fact, in that interval, : nx 1 te 0 < 50 (1) < ps 2 = yu The limiting curve is therefore the stretch 1 << x << 2 on the axis of #. The n'® curve of approximation lies above this stretch and, by the above inequality, at a : 1 Sa distance of less than = from the limiting curve, throughout the whole of the interval 1 < x <2. For large »’s, the distance all along the curve is therefore = Nor very small. : ee ey In this case, therefore, matters are much as we should expect; the position is “entirely altered if we consider the same series in the interval 0 <2 <1. We still. have lims, (x) =0 at every point of this interval4 so that the limiting cuive is the corresponding portion of the x-axis. But in this case the nt? approximation curve no longer lies close to the limiting curve through- : 1 out the interval, for amy n (however large). For Bey We have always 1 iy : : 5 () = so that, for every =, the approximation curve in the interval from 0 to 1 has a hump of height Lo The graph of the curve y =s, (x) has the following appearance: : Fig. 4. t In. fact, for 2 >0 we have 0 for x=0, s, (x) =0 even permanently. § 46. Uniform convergence. - 331 The curve y = s,, (x), however, corresponds more nearly to the following graph: Fig. 5. For larger m's, the hump in question — without diminishing in height — be- comes compressed nearer and nearer to the ordinate-axis. The approximation curve springs more and more steeply upwards from the origin to the height a, which it attains for z=, only to drop down again almost as rapidly to within a very small distance of the z-axis. The beginner, to whom this phenomenon will appear very odd, should take care to get it quite clear in his mind that the ordinates of the approxima- tion curves do nevertheless, for every fixed x, ultimately shrink up to the point -. ou the x-axis, so that we do have, for every fixed u, lims, (3) =0. li zis. given a fixed value (however small), the disturbing hump of the curve y= s, (2) will ultimately, i. e. for sufficiently large #’s, be situated entirely fo the left of the ordinate through x (though still fo the right of the y-axis) and on this ordinate the curve will again have already dropped very close to the z-axis?. Therefore the convergence of our series will be called wniform in the interval 1 <2 <2, but not in the interval 0 <2 < 1. We now proceed to the abstract formulation: Suppose 2'f, (x) possesses an interval of convergence J; it is convergent for every indiv- idual point of J, for instance at x = x; this means that if we write Fa)==3 (2) +7, (x) and assume & > 0 arbitrarily given, there is a number x, such that, for every n > Ng» | 7, (xo) | =e. Of course the number #,, as was already emphasized (v. 10, rem. 3), de- pends on the choice of &. But ny now depends on the choice of x, also. In fact for some points of J the series will in general converge more 5 If we take, say, x = and » =1000000, the abscissa of the highest 1 1000 point of the hump is 1 100000g° 2nd at our point x. the curve has already ; dropped to a height <7o° 332 Chapter XI. Series of variable terms. rapidiy than for others®. By analogy with 10, 3, we shall therefore write ny = n,(e,&,); or more simply, dispensing with the index 0 and with the special emphasis on the dependence on &, we shall say: Given ¢ > 0 and given « in the interval J, a number x (r) can always be assigned, such that for every n > n(x), 7, (@)]| N, but also for every x in J . We also say that the remainders 7, (x) tend uniformly to 0 in J. Illustrations and Examples. 1. Uniformity of convergence invariably concerns a whole interval, never an isolated point. 2. A series J f, (x) convergent in an interval J does not necessarily con- verge uniformly in any sub-interval of J. 3. If the power series Xa, (x — x," has the positive radius 7 and if 0

0, choose N= N(¢) so that for every n> N | npr |-0" t+ | apse] 0" T+. N, we certainly have |#, (x)| N and 0 1. ; +1 If, for instance, we choosen >> 4 and >> N, and so large that (1 —_ 1) = = ’ ay ; 1 etl 1 1 . — this is possible, as 1 PE => = Ty the remainder, for such a value of # and z= (1 =) J 18 1 7, (@) = afr LY >t Log ed, 7. The above clears up the meaning of the statement: Xf, (x) is not uniformly convergent in a portion J’ of its interval of convergence. A special value of ¢, say the value g, >> 0, exists, such that an index = greater than any assigned N may be found, so that the inequality |.%, (x) | 0, we draw the two curves y = F (x) + ¢, the approximation curves y = s, (x) will ultimately, for every sufficiently large n, come to lie entirely within the strip bounded by the two curves. 12 534 Chapter XI. Series of variable terms. 9. The distinction between uniform and non-uniform convergence, and the great significance of the former in the theory of infinite series, were first re- cognized (almost simultaneously) by Pi. L. v. Seidel (Abh. d. Minch. Akad., p. 383, 1848) and by G. G. Stokes (Transactions of the Cambridge Phil. Soc., Vol. 8, p. 533. 1848). It appears, however, from a paper by K. Weierstrass, un- published till 1894 (Werke, Vol. 1, p. 67), that the latter must have drawn the distinction as early as 1841. The concept of uniform convergence did not become common property till much later, chiefly through the lectures of - Weierstrass. Other forms of the definition of uniform convergence. o3™ form. JX'f, (x) is said to be uniformly convergent in J if, in whatever way we may choose the sequence (x,)° in the interval J, the corresponding remainders Tn (0) invariably form a null sequence’, : We can verify as follows that this definition is equivalent to the preceding: a) Suppose that the conditions of the 27d form of the definition are fulfilled. Then, given &, we can always determine N so that | 7, (z)| < & for every n> N and every x in J’; in particular lr, (@ |< efor every n>N; | hence 7, (z,)—0. 3 b) Suppose, conversely, that the conditions of the 3 form are ful- filled. Thus for every (x) belonging to J’, #,(x,)— 0. The conditions of the 22d form must then be satisfied also. In fact, if this were not the case, — if a number N = N(¢) with the properties formulated there ] did not exist for every € > 0, — this would imply that for some special €, say ¢ =g¢,, no number N had these properties; above any number N, however large, there would be at least one other index # such that, for some suitable point x =x, in J’, |7,(z,)| => ¢, (cf. our last example, in which ¢g=1 and z =2, =1 = Let n, be an index such that |7,, (x,)| => ¢,- Above un, there would be another index #,, such that | 7,, (2,,) | = ¢, for a suitable corresponding point ,,, and so on. 9 The sequence need not converge, but may occupy any position in J’. 10 Should each of the functions | 7, (z)| attain a maximum in J’, we may choose z, in particular so that |#,(x,)|= Max], (z)|; our definition thus = takes the special form: Xf, (x) ¢s said fo be uniformly convergent in J’ if the maxima Max |v, (x) | in J' form a null sequence. If the function |7, (x) | does not attain a maximum in J’, it has, however, a definite upper bound u,. We may also formulate the definition in the general form: : OForm 8a. Xf, (a) is sad to be umformly convergent in J’ if u, — 0. (Proof?) = § 46. Uniform convergence. 3385 We can choose (x) in J’ ‘so that the points #,,,%,,... belong to (x,), in which case . n 7. (2,) will certainly not form a null sequence, contrary to hypothesis. Our assump- tion that the conditions of the 27d form could not be fulfilled is inadmissible; the 37 form of the definition is completely equivalent to the 2nd, In the previous forms of the definition, it was always the remainder _ of the series which we estimated, the series being already assumed to converge. By using portions of the series instead of infinite re- mainders (v. 81) the definition of uniform convergence may be stated so as to include that of convergence. We obtain the following definition: 04th form. A series Xf, (x) is said to be uniformly convergent in the interval J’ if, given ¢ > 0, we can assign a number N = Ne) independent of x, such that |fas1(0) + Fare (@) +--+ -F Fain (x) | <& for every n> N, every E21 and every xz in J'. Or we may finally express it in the Obth form. A series 2'f, (x) ts said to be uniformly convergent in the interval J if, when positive integers k,, ky, ky, ... and points Ty, Xp, Xg, ... Of J ave chosen arbitrarily, the quantities [fon (xn) + frie (ocr) = 2 + fnin, (.)] invariably form a null sequence’. Further Examples and Illustrations. 1. The student should examine afresh the behaviour of the series 3, (x), with 192. nx 1+ n%a? a) in the interval 1 <2 < 2, b) in the interval 0 <2 <1 (cf. the considerations on pp. 330—1). Sn (x) = 2. For the series 1+ @—1)4@—a)+--- + (a" — a" 1) + 11 By 51, we might even write fr, +1@) + +1 tr, (2:)] for the above, where the »,’s are any integers tending to 400. Exactly as in 81, we may speak of a sequence of portions, except that here we may substitute a different value of z in each portion. The statement we then obtain is: A sevies 2'f, (x) is said to be wniformly convergent in J' if every sequence of portions of the series forms a null sequence. Similarly: A sequence of functions Su (%) is said to be uniformly convergent in J' if every diffevence-sequence is a null Sequence. 336 Chapter XI. Series of variable terms. we have obviously s, (x) =a". The series accordingly converges in the inter- val J: —1 1. 13 In spite of this, it is easy to see that for every fived x (in 0 <<» <1) the values s, (x) diminish to 0 as » increases, so that the abrupt rise to the height 1 occurs to the right of x, however near x may be taken to + 1, provid- ed only that # is chosen sufficiently large. ; < § 46. Uniform convergence. 337 appearance to those in Figs. 4 and 5, with this modification, that the height of the hump now increases indefinitely with #; this is because 4. We must emphasize particularly that uniform convergence does not require each of the functions fj (x) to be individually bounded. The series 1 z +14 2x+2%+-.--, for instance, is uniformly convergent in 0 0 and a number G > 0 such that |g, (x)| < G for every x in J and every n > m. (I e.: A series which still converges uniformly when its terms ave taken wn absolute value may be multiplied term by term by any functions all but a finite number of which, at most, are uniformly bounded wm J.) : : 1. : 14 The point for which x = — is actually the maximum point of the curve n ¥ =s, (x), as may be inferred from s/ = (rn — n? 2?) airy. 193. 338 : Chapter XI. Series of variable terms, O Theorem 4. i 2f, (x) converges uniformly in J, then for a suitable m the functions fpi1(%), fum+2 (x), ... are uniformly bounded in J and converge uniformly to O. O Theorem 5. If the functions g, (x) converge uniformly to 0 in J, so do the functions y, (%)g, (x), where the functions y, (x) are any functions defined in J and — with the possible exception of a finite number of them — uniformly bounded in J. We may give as a model the proofs of Theorems 3 and 4: Proof of Theorem 3. By hypothesis, given ¢> 0, we can de termine n, > m so that for every n > un, and every zx in J, FACIE STARE) TEE For the same #’s and 2's we then have J Gibii Fi S Gn la rr Cl i ed xe: This proves all that was required. Proof of Theorem 4. By hypothesis, there exists an sm such that, for every n> m and every x in J, |7,(#)| <1. Hence for n >m and every » in J, L@=17_ 0) 0, Kr, in] x, which proves the first part of the theorem. If we now choose n, > m so that for every n >#n, and every x in J, |7,(x)| < le, (¢ being pre viously assigned) the second part follows in quite a similar way. § 47. Passage to the limit term by term. Whereas we saw on pp. 328—9 that the fundamental properties of the functions f, (x) do not in general hold for the function F(z) represented by Jf, (x), we shall now show that, roughly speaking, this will be the case when the series is uniformly convergent’). 8 We first give the following simple theorem, which becomes parti- cularly important in applications: 3 © Theorem 1. If the series 2f, (x) is uniformly convergent in an : v 3 1 interval and if its terms f, (x) are continuous at a point x, of this interval, the function F(x) represented by the series is also continuous at this point. 1» We may, howeyer, mention at once that uniform convergence still only represents a sufficient @@ndition in the following theorems and is not in general necessary. % >, § 47. Passage to the limit term by term. 339 Proof. Given & > 0, we have (in accordance with § 19, Def. 6b) to show that a number § = d(g) > 0 exists such that | F(z) — Fm) By the assumed fact of uniform convergence, we can choose n =m so large that, for every x in the interval, |7, (2) J =. Then | F@) — F(@)| <5, (%) — 8 (®0)| +5 The integer m being thus determined, s, (x) is the sum of a fixed number of functions continuous at x, and is therefore (by § 19, Theorem 3) itself continuous at ,. We can accordingly choose J so small that for every x in the interval for which | —z,| < J, we have | $m (&) — $1 (%0)| < ++ For the same 2's we then have |F (2) — F(z)| xg n= 0 x—>axy In this form it appears as a special case of the following much more elaborate theorem: © Theorem 2. We assume that the series F(x) = FA (x) is uni- 194. formly convergent in the open interval x,...x, 1° pnd hat the limit, when x “approaches x, from the interior of the interval 17, Tr Zo 16 3, may be > or a, then converges and lim F (x), when x — x, in n=0 the above manner, exists. Moreover, if we write 2a, = A, we have lim F(z) =A4 XT> Xo or, otherwise, lim (37, (x) = 3 (lim, (2). a—>»xg n=0 n=0 x>2z, (The latter form is expressed shortly by saying: In the case of uni- form convergence, we may proceed to the limit term by term.) Proof. Given ¢ > 0, first choose n,, (v. 4th form of the defini- tion 191) so that for every # > n,, every > 1 and every z in our interval, in ®t D] = &, Let us for the moment keep n and % fixed, and make x—2,. By § 19, Theorem 1a, it follows that 2,4: Pe] Se And this is true for every n >n, and every k>1. Hence Za, is convergent. Let us denote the partial sums of this series by 4 and its sum by 4. Itis easy to see now that F(z)— A. If, for a given, n, is determined so that, for every # > n,, we not only have 7, (0) < 3» but also 4-4, <3, then, for a (fixed) m > n,, : | F(z) — 4] = |(s,, (* — (4 — A4.)+ 7.2 | Ls, (@) — 4, + 5+ As z— x, involves s, (¥)— A4,,, we can determine J so that js, (0) — 4, | <2 for every x belonging to the interval, such that 0 < |x — | < 4. For these «2's, we then also have |F(z)— 4| 0, we determine 5, so large that for every n> n, and every in a...b, g - 7, 0, we may (by § 19, theorem 11) enclose the possible points of dis- continuity of the integrable function f(x) in intervals of total length <4 similarly those of f, (x) in intervals of total length <2 and generally those of le (x) in intervals of total length < gor - All the points of our interval at which any of our functions are discontinuous are thus enclosed in intervals of total length cosn'zy, - n=1 which is divergent for every z®. — Even if a series converges uniformly for every z, as for instance the series Br Nye 2 n=1 (cf. Example 5,191, 2), the position is no better, since on differentiating term by term we obtain A a series which diverges e. g. for x =0. The theorem on term-by-term differentiation must accordingly be of a different stamp. It runs as follows: 196. Theorem 4. Given a series > f., (@)®° whose terms are differen- n=0 tiable in the interval J=a...b, (a 4. (x), n=0 deduced from it by differentiating term by term, converges uniformly in J, then so does the given series, provided it converges at least at one point of J. Further, if F(x) and ¢ (%) are the functions represented by the two series, F(x) is differentiable, and we have F (x) = ¢ (x). In other words, with the given hypotheses, the series may be differ- entiated term by term. 19 The formulae established on p. 8357 give, for every x + 2k x, 1 sin (» + 7) z >t cosz+cos2x+.--fcosSnE=—"""——. . 2 sing 20 As regards the convergence of the series, no assumption is made in the first instance. ! § 47. Passage to the limit term by term. 343 Proof. a) Let ¢ denote a point of J (existent by hypothesis) for which Xf, (c) converges. By the first mean value theorem of the differential calculus (§ 19, theorem 8) Sher -e-9: ILO, where & denotes a suitable point between x and ¢. Given ¢ > 0, we can, by hypothesis, choose n, so that for every n > n,, every k > 1, and every x in J, n+k = = r=n+1 —-_—d Under the same conditions, we therefore have EF h@— 10) This shows that 2(f, (x) — f, (c)), and hence Xf, () itself, is uniformly convergent in the whole interval J and accordingly represents a de- finite function F(z) in that interval. 'b) Now let x, be a special point of J and write Ele =I (@) == 0 (1), (r=0, Et 2.0) These functions are defined Wi every hz 0 for which x, 4h belongs to J. As above, we may write n+k n+k 2 nW=2 5 E400 00 and if 0

0=[F() dx. y=140 0 0 Thus term-by-term integration leads to the correct result. In the case of the 1 series 192,3, however, in which we also have JF) dx =0, term-by-term 0 integration gives, on the contrary, 1 —~1n fro) de=]s@dz=l—e * ‘1. 0 Sn In this case, therefore, term-by-term integration is not allowed. § 48. Tests of uniform convergence. Now that we are acquainted with the meaning of the concept of uniform convergence, we shall naturally inquire how we can de- termine whether a given series does or does not converge uniformly in the whole or a part of its interval of convergence. However difficult it may be — and we know it often is so — to determine the mere convergence of a given series, the difficulties will of course be considerably enhanced when the question of uniform convergence N § 48. Tests of uniform convergence. 345 is approached. The test which is the most important for applications, be- cause it is the easiest to handle, is the following: O Weierstrass’ test. If each of the functions f, (x) is defined and 197. bounded in the interval J, — say If,®)|L 7, throughout J — and if the series 2p, (of positive terms) converges, the series 2 f, (x) converges uniformly in J. Proof. If the sequence (x,) is chosen arbitrarily in J, we have | fos1(2,) Yr: hal) sor borin BB Vues Toda Hr rch Pasty By S81, 2, the right hand side —0 when #— oo; hence so does the left. By 191, 5 form, 2'f, (x) is therefore uniformly conver- gent in J. Examples. 1. In the example 191, 3 we have already made use of the substance of Weierstrass’ test. 1 ; ; ; 2. The harmonic series STE which converges for x >1, is uniformly convergent on the semi-axis x => 1-4 J, where J is aly positive number. In fact, for such «’s, 1 1 WE | ST =e n plt where X'y, converges. This proves the statement. The function represented by the harmonic series — known as Riemann’s {-function and denoted by ¢ (zx) — is therefore certainly continuous for every 1.2 3. Differentiating the harmonic series term by term, we deduce the series @ logmn n=1 n This again is uniformly convergent in # => 146 > 1. In fact, for every suffi- log n ciently large n, Se <1 (by 88,5); for these »’s and for every z=1 +, we then have log n 1 log n 1 ne |= Ate 5 ira Ir Riemann’s -function is accordingly differentiable for every x > 1. 4. If Xa, converges absolutely, the series = 4, COS and >a, sinny ~ are uniformly convergent for every xz, since e. g. |a,cosnz |< a, =y,. These ~ series accordingly define functions continuous everywhere. : In spite of its great practical importance, Weierstrass’ test will necessarily be applicable only to a restricted class of series, since it 2 In fact, if we consider a special x > 1, we can always assume § > 0 chosen so that x > 14-4. 198. 346 Chapter XI. Series of variable terms. requires in particular that the series investigated should converge absolutely. When this is not the case, we have to make use of more delicate tests, which we construct by analogy with those of § 43. The most powerful means for the purpose is again Abel's partial summation formula. On lines quite similar to those already followed, we first ob- tain from it the OTheorem. A series of the form > a, (x)-b, (x) certainly converges uniformly in the interval J, if, in E% 1) Fan, b,.,) is uniformly convergent (as a series) and 2) (4,-b,.,) ts uniformly convergent (as a sequence) Here the functions A, = A, (x) denote the partial sums of a, (z). Proof. As formerly — we have merely to interpret the quant ities @,, b, and 4, as no longer numbers, but functions of x — we first have n+k n+k > a,b, = = 4,0, D010 Eh (dori Vusnsr = A4,0,..) v=n+1 r=n+1 Letting « and % vary in any manner with n, we have on the left a sequence of portions nt+kn a,(x,)-b, (,) r=n+1 of the series 2a, b,, and on the right the corresponding ene relative to the series 34 (b, — b,,,), and a difference-sequence of the sequence (4,+b,,,)- Since by hypothesis the latter sequences always tend to 0 (v. 191, 5th form), it follows that so does the sequence on the left. This (again by 191, 5), proves the statement. Exactly as in § 43, the above theorem, which is still very general in character, leads to the following more special, but more easily man- ageable tests 23: 8 ©1. Abel's test. Za, (x)-b, (x) is uniformly convergent in J, if Za, (x) converges uniformly in J, if further, for every fixed value of x, the numbers b,(x) form a real monotone sequence and if, for 2 g.,b,, 4, are now always functions of x defined in the interval J; only i for brevity we often leave the variable zz unmentioned. — For the notion of the uniform convergence of a sequence of funciions cf. 190, 4. E 28 For simplicity’s sake, we name these criteria after the corresponding 3 ones for constant terms. — Cf. p. 315, footnote 8. a § 48. Tests of unitorm convergence. 847 every m and every x in J, the functions b, (x) are less in absolute value than one and the same number K?*. Proof. Let us denote by e (x) the remainder corresponding to the partial sum 4 (2); i. e. 2a, (®) = 4, (x) +a, (x). In the formula n= of Abel's partial summation, we may (by the supplement 183) sub- stitute — e¢, for 4,, and we obtain n+k n+k = a,b, =a e,-(b, ae bis) ak Within = 0,11) r=n+1 r=n+1 . it therefore again suffices to show that both Xe, (b, —b,,,) and (¢,-b,.,) converge uniformly in J. However, the ¢, (2s, as remainders of a uniformly convergent series, tend uniformly to 0 and the b,(z)’s remain < K in absolute value for every x in J; it follows that (e, +b, ,) also converges uniformly to 0 in J. On the other hand, if we con sider the portions n+k Te = - «, (2) (0, (2) b, +1), r=n+1 we can easily show that these tend uniformly to 0 in J, — thereby completing the proof of the uniform convergence in J of the series under discussion. In fact, if @, denotes the upper bound of « (x) in J, — 0 (v. form 8a). Thus if ¢, 1s the largest of the numbers @ ., this ¢, also —0 and e, ntl? «o nT" n+k [2 = oe 3) b, oe b,41 | = 810,404 Eh Darna) = 2K:¢, involves the fact that 7, — 0 uniformly in J. ©2. Dirichlet’s test. > a, (x)-b, (x) is uniformly convergent in J, n=0 > if the partial sums of the series 2 a (x) are uniformly bounded in J2* and if the functions b, (x) converge uniformly to O in J, the conver- gence being monotone for every fixed x. Proof. The hypotheses and 192, 5 immediately involve the uniform convergence (again to 0) of (4,-b, ,,). If, further, K’ denotes # The b, (z)'s form, for a fixed x, a sequence of numbers b, (x), b, (sive for a fixed n, however, b, (z) is a function of x, defined in J. The above as- sumption may, then, be expressed as follows: All the sequences, for the various values of x, shall be uniformly bounded with regard to all these values of x; in other words, each one is bounded and there is a number K which is simultaneously a bound above for them all. Or again: All the functions defined in J for the various values of » shall be uniformly bounded with regard to all these values of »; i. e. each function is bounded, and a number K exists which simultancously exceeds them ail in absolute value. 348 Chapter XI, Series of variable terms. a number greater than all the | 4, (2)['s for every x, we have n+k , n+k ; S4,:0,=05,9) 58 2 = 10,0, 058 0, erin Deka rv=n+1 r=n+1 J In whatever way x and k may depend on #, the right hand side will tend to O by the hypotheses, hence also the left. This proves the uni- form convergence in J of the series under consideration. The monotony of the convergence of b, (x) for fixed x has only been used in each of these tests to enable us to obtain convenient upper estimations of the portions 2'|b, — b,,, |. By slightly modifying the hypotheses with the same end in view, we obtain 03. Two tests of du Bois-Reymond and Dedekind. a) The series 2 a, (x)-b, (x) is uniformly convergent in J, if both Sa, and 2 |b, — b,41| converge uniformly in J and if, at the same time, the functions b, (x) are uniformly bounded in J. Proof. We use the transformation n+k ntk - 7,0, = — > eb — Dei) (0, +30, 0040 =~, Db.) v=n+1 r=n+1 As the remainders «, (x) now converge uniformly to 0, we have, for every » >a, say, and every ® in J, |o (vr) <1. Hence for every n=>m, n+k n+k sod 0) = 2 Jo 0,541 r=n+1 r=n+1 the expression on the right — even if # and 2 are made to depend on 7, in any manner — now tends to O as x increases, hence so does the expression on the left. That «,-b,,, tends uniformly to 0 in J follows, by 192, 5, from the fact that « (x) does and that the b, (x)’s are uniformly bounded in J. b) The series 2 a, (x)-b,(x) is uniformly convergent in J if only the second of the lwo series Xa, and 2 |b, —b, | converges uni- formly in J, provided that the former has uniformly bounded partial sums and the functions b, (x)— 0 uniformly in J. Proof. From the hypotheses, it again follows at once that 4b, ,, converges uniformly (to 0) in J. Further, if K’ once more denotes a number greater than all the | 4, (x)[s for every «x, n+k | ; n+k 2 4,0, ~ +1) =K 2 | b, byt1l> v=n+1 y=n-+ whence, on account of our present hypotheses, the uniform conver- gence in J of the series 2 4, (b, — b,,,) may at once be inferred. \ § 48. Tests of uniform convergence. : 349 > Examples and Illustrations. 1. In applications, one or other of the two functions a, (x) and b, (¥) 199. will often reduce to a constant, for-every =; it will usually be the former. Now a series of constant terms Xa, must, if it converges, of course be re- garded as uniformly convergent in every intevval, for, its terms being independent of x, so are its portions, and any upper estimation valid for the latter is valid ipso facto for every x. Similarly the partial sums of a series of constant terms Xa,, if bounded, must be accounted uniformly bounded in every interval 2. Let (a,) be a sequence of numbers with Xa, convergent, and let bn (¥) =z". The series Ja, x is uniformly convergent in 0 << x <1, for the conditions of Abel's test are fulfilled in this interval. In fact, Xa,, as re- marked in 1., is uniformly convergent; further, for every fixved x in the inter- val, (z") is monotone and |z"| <1. — By the theorem 194 on term-by-term passage to the limit, we may therefore conclude that im (Se,2)=2 (lim «¢,27), ie z->1-0 r=>1-—0 =3 Ay « This gives a fresh proof of Abel's limit theorem 100. 3. The functions b, = also form a sequence bounded uniformly in J (namely, again <1), and monotone for every fixed xz. Hence, as above, we deduce that : lim 'X7 in > apy > 0 BF if Xa, denotes a convergent series of constant terms. (Abel's limit theorem or Duvichlet series.) 4. Let a, (x) =cosnxz or =sinnz, and n=, a> 0. The series n 2 > coSnL ® sinnz > ane Sted or LY oR {>0), n=1 n=1 n=l then satisfy the conditions of Divichlet’s test in every interval of the form d= x<22—0%, where 0 denotes a positive number < sx. In fact, by 185,5, the partial sums of Xa, (x) are uniformly bounded in the interval i may take Le and b, (xr) tends monotonely to 0, sin 5-4 — uniformly, because b, does not depend on z. — If (b,) denotes any monotone null sequence, it follows for the same reason that 2b,cosnz and Zh, sinnz are uniformly convergent in the same intervals (cf. 185, 5). — All these series accordingly represent functions which are defined and continuous? for every ? Or in intervals obtained from the above by displacement through an integral multiple of 2 a. . ; * Every fixed x += 2 k # may indeed be regarded as belonging to an inter- ~ val of the above form, if d is suitably chosen (cf. p. 843, example 1, and p. 345, footnote). 350 Chapter XI. Series ot variable terms. tv == 2k x. Whether the continuity subsists at the excluded points z=2%k x we cannot at once determine, — not even in the case of the series 2 b, sin nz, although it certainly converges at these points (cf. 216, 4). § 49. Fourier series. A. Euler’s formulae. Among the fields to which we may apply the considerations developed in the preceding sections, one of the most important, and also one of the most interesting in itself, is provided by the theory of Fourier series, and more generally by that of trigonometrical series, into which we now propose to enter?” By a trigonometrical series is meant any series of the form 1 = : 5% + 2 (a, cosu .b, niem, n= ~ with constant a, and b, 8. If such a series converges in an interval of the form ¢ 0: Som, a> 1; ec n=1 n=1 n We have never been in a position, so far, to determine the sum of any of these series for all values of x. It will appear very soon, how- ever, that trigonometrical series are capable of representing the most curious types of functions — such as one would not have ventured to call functions at all in Euler's time, as they may exhibit discon- tinuities and irregularities of the most complicated description, so that they seem rather to represent a patchwork of several functions than to form one individual function. 27 More or less detailed and extensive accounts of the theory are to be found in most of the larger text books on the differential calculus (in parti- cular, that referred to on p. 2, by H.v. Mangoldt, Vol. 3, 2nd ed., Part. 8, 1920). For separate accounts, we may refer to H. Lebesgue, Legons sur les séries tri- gonométriques, Paris 1906, and to the particularly elementary Introduction to the theory of Fourier’s series, by M. Bécher, Annals of Math. (2), Vol. 7, pp- S1—152. 1906. A particularly detailed account of the theory is given by E. W. Hobson, The theory of functions of a real variable and the theory of ; Fourier series, Cambridge 1907. 3 Ne TT ean SR I 28 Tt is only for reasons of convenience that —- a, is written instead of 2,. 2 § 49. Fourier series. — A. Euler’s formulae. 851 Thus we shall see later on (p. 375), that e.g. ain =0 for x=kax, (k=0,+1, +2,...), but = n {erina—s for 2ka ~& [ren Stopes A) sim — 2 i. e. by the above remark with regard to f Sg sin @nt+1) + — fie 0 — 1): i A sin — and we accordingly obtain sin @n +41) + 1 [tents : 2. 5s, (7) = 7 sin 7 Substituting 2 ¢ for #, we are ultimately led to the formula TT 2 2 2% —2¢) sin (2 1) 1 Sul) =2 [Lt LLL Ian 08 yy, re This is Dirichlet’s integral*', by which the partial sums of the Fourier series generated by f(x) may be expressed. We may therefore state, as our first important result, the theorem: Theorem 2. In order that the Fourier series generated by a fumnc- tion f(x), integrable (hence bounded) and periodic with period 2m, may 41 We designate as Dirichlet’s integrals all integrals of either of the two forms a sink: sin k¢ fe ©) 3 or IEC : dt. 0 0 =~ 360 Chapter XI. Series of variable terms. converge at a point x,, it is necessary and sufficient that Dirichlet’s integral 2 = f@+2)+f@—20 sn@ntlr 2 2 sin ¢ should tend to a (finite) limit as m — oo. This limit is then the sum of the Fourier series at the point x. Let us denote this sum by s(z,). The second question (p. 355), concerning the sum of the Fourier series, when convergent, may be included in our present considerations and our result may be put in a form still more advantageous in the sequel, by expressing the quantity s(x,) in the form of a Dirichlet integral also. As sin (27 +4 18 5 5 + cost 4-cos 2st - + cosnt= —— 2sing we have : : 2rsin(@n+1)— 2 FE, 5 2sin 203. or, effecting the same transformations as before with the general integral, sin(2n+1)¢ (b) zy on di=2 5 4 0 Multiplying by s(z,), we finally obtain, by subtraction from 202, wo] q TT 02 0o— 2 in(2n+1 oo) sl) L [LEO m2 _ nat Dey sin? 0 Our preceding theorem may now be expressed as follows: Theorem 2a. In order that the Fourier series generated by a function f(x), integrable and periodic with period 27, should converge to the sum s(x,) at the point xy, it is necessary and sufficient that, as n — 4-00, Dirichlet’s integral sin sin(2n+41)¢ : Foe 2) Cr EDL gy 42 This formula may also be obtained from 202, by substituting f)=1; ; this gives a, = 2 and, for every = 21 8,=0,=0,1.e.5,(;)=1 for every #3 and every x. a : § 49. Fourier series. — B. Dirichlet’s integral. 361 should tend to 0, where for brevity we have put +22) + — 21 FEET Although this theorem by no means solves questions 1 and 2 in such a manner that the answer in given concrete cases lies ready to hand, yet it furnishes an entirely new method of attack for their solution. Indeed the same may be said with regard to the third of the questions proposed on p. 355, for theorem 2a may at once be modified to the following: Theorem 3. On the assumption that the partial sums s, (x) con- verge to s(x) at every point of the interval « 0, we can assign I= N(e) so that 2 free ,sin@nt1)t y sin ¢ for every n> N and AE Times e0, as its r orand 3 is never negative. On the other hand, it is 27 25 27 Jf@Pdt—23[a, [ fH) cosvtdaf] — 23 [b, [ f(t) sinve dt] 0 0 0 +] [2 (a,cosvt +b, sinwi)]?dt 0 2x7 =[[fOPdt—2a2a] — 2230) +nZal +a 3b] 0 x ~ [irera — = Za +b), where each summation is extended from » — 1 to » = x. Since this expression is non-negative, we have 2m Sart (rapa. 362 Chapter XI. Series of variable terms. Thus the partial sums of the series (of positive terms) in question are bounded and the series is convergent, as asserted. The above contains in particular Theorem 5. The Fourier constants (a,) and (b,) of an integrable function form a null sequence. From this, we may deduce quite simply the further Theorem 6. If y(t) is integrable in the interval a 0 — the new integral 3 2 sin (2n+1)¢ 2 {o; myn 0 tends to 0 as » increases. Now the latter integral only involves the values of f(x, + 2f)in0L1 0 may be assumed arbitrarily small, this remarkable result contains at the same time the following Theorem 7. (Riemann’s theorem.*®) The behaviour of the 204. Fourier series of f(x) at the point x, depends only on the values of f(x) in the neighbourhood of x,. This neighbourhood may be as- sumed as small as we please. In order to illustrate this peculiar theorem, we may mention the following consequence of it: Consider all possible functions f(x) (inte- grable in 0...2z) which coincide at a point x, of the interval 0...2x and in some neighbourhood of this point, however small, possibly varying with the particular function. Then the Fourier series of all these functions — however much they may differ outside the neigh- bourhood in question — must, at a, itself, either all converge or all diverge, and in the former case they have the same sum s(z,) (which may or may not be equal to f(x,)). After inserting these remarks, we proceed to reformulate the criterion obtained above, which we may henceforth substitute for theorem 2: Theorem 8. The necessary and sufficient condition for the Fourier series of f(x) to converge at x, to the sum s(x), is that for an ar- ~ bitrarily chosen positive 6 < Zz Dirichlet’s integral 8 2 sin (2n-+1)¢ fy {; ny Ble, sin ¢ 0 should tend to 0 as n increases*®. 4% Uber die Darstellbarkeit einer Funktion durch eine trigonometrische Reihe, Hab.-Schrift, Gottingen 1854 (Werke, 20d ed. p. 227). 46 As regards uniformity of convergence, we can assert nothing straight away, since we are ignorant as to whether the integral (c) above considered, which tends to 0 as = increases, for every fixed x,, will do so wumiformly for every z of a specified interval on the z-axis. Actually this is the case, but we do not propose to enter into the question further. 364 : Chapter XI. Series of variable terms. There is no difficulty in showing that the denominator sin in the last integrand may be replaced by ¢{. In fact the difference be- tween the original integral and the one so obtained, i. e. the integral 8 2 1 Tl, 2 fo ,) a 2]-sin(2n + 1)¢- dt, 0 automatically tends to 0 as » increases, by theorem 6, — because os — 3 is continuous and bounded #7, and hence integrable, in 0 < i< 9. Thus we may finally state: Theorem 9. The necessary and sufficient condition for the Fourier series of a function f(x), periodic with the period 2 zt and integrable over 0...27, to converge to s(x,) at the point x,, is that for an arbitrarily chosen positive s(< 2) the sequence of the values of the integral 8 2 yt; my SLl2 pie, 0 forms a null sequence. Here @ (3; x,) has the same meaning as in theorem 2a. In another form, thecondition is that, given ¢ > 0, we can assign 0 <3 and N > 0, so that, for every n > N, 2 0 uy npr, < ett C. Conditions of convergence. Our preliminary investigations have prospered so far that the first two questions of p. 3556 may now be attacked directly. By the above, these are completely reduced to the following problem: : Given a function @(f), integrable in 0 +® E>+» where @, denotes the (right hand) limiting value lim ¢(f), which cer- tainly exists with the assumptions made E340 Proof. 1) Inthe first place, The existence of this limit, i. e. the convergence of the improper in- tegral, follows simply from the fact that, given ¢ > 0, and any two values 2 and 2” both > > we have (by § 19, theorem 26) z' z' sin ¢ cos 11%" [= dies [- | ~ Z d xz z fora) ik 5st 2, 5 There is no plitionion in observing that it would suffice for % to ~ tend to 4 oo through odd integral values. : M In fact, as ¢ (¢) is integrable, it is certainly bounded, and by hypothesis it is monotone in 0 <¢ 1 IX i =f (i — +) sin (2 » +1)¢-d¢ 0 : (cf. the developments on p. 364) form a null Seouonet, by theorem 6. Accordingly we also have , : ; in (2 or” hn 4 — [= SL far 3. 0 Since, however (v. § 19, theorem oo n+1) ay 2 0 this implies that the above-named limit has the value = | Q 2 - 2) By 1), a constant K’ exists such that fe at] < er for every # > 0, and therefore a constant K(= 2 K’) exists such that b [5a 0 and choose a positive ¢' Fk. Further, &' o' (4) I 2 [lp sh 2g pl LP [25 ar ” 4 J, 0 0 For the second of these two quantities, we have ko’ ® sin ¢ 2 sin ¢ selina nme 0 0 and we may accordingly choose k, > k’ so large that & |i” —nl k,. For J”, the first of the two quantities on the right of (d), we use the second mean value theorem of the integral calculus § 19, theorem ” which gives, for a suitable non-negative ¢” <¢/, oO ~2fip0- Eg = 2 [p(8) — gu): [2a 44 ko! The latter integral — to, and therefore remains << K in absol- £8” ute value, by 2). Accordingly ” ¥ k <2 7 w Kad ~ Combining the three results of this by means of Ix Ea iL at I ak Tir 4 J we see that, given & > 0, we can choose k, so that, for every % > &,, Va— wl S| =F HE 53-5 Thereby the statement is completely established. 2. Dini’s rule. If lim (tf) = @, exists, and if for every positive t->+0 Tt <0, the integrals j 8 Ju = Pol 74 remain less than a fixed positive number®®, then lim J. exists and Po- 4 k>+ ® 92 More shortly: If the integral flea dt, which is improper at 0, has a meaning. 368 Chapter XI. Series of variable terms. Proof. When 7 decreases to 0, the above integral increases mono: tonely but remains bounded; it therefore tends to a definite limit as 7— 0, which we denote for brevity by ds [let nly, J t Given & > 0, we may choose a positive 0’ < J so small that 5 lo ®)—a g fleo=niy zs 0 Writing, as in the previous proof, 8 2 sin k¢ 7 ar 2 fy t) — dit and I. es J 1 Sn z 0 the difference (J, — J) tends to 0, by theorem 6, and we may choose k so large that | J, — J] <7 for every k > k’. Further, as we saw before, with a suitable choice of k, > %’ we also have 0 2 sink ¢ : | 1) — @o| = Lo 1 dt — @, <3 for every k > k,. Finally, é' 8 ; 2 sinkt D—o 127 1= [2 [lp y= pd tar] < [Lem 0 0 i. e. when ¢' is suitably chosen, | J,” | also remains < 3 Thus, pre cisely as before, we conclude that, for every k > &, | Je — ®ol < es which proves the validity of Dini’s rule. We may easily deduce from it the two following conditions. 3. Lipschitz’s rule. If two positive numbers A and « exist, such that lo) — po] < A-t2 for every t in 0 +0 § 49. Fourier series. — C. Conditions of convergence. 369 so that for every positive 7 << § the former integral remains less than a fixed number and in consequence of Dini’s rule J, — @,, as required. 4th rule. If ¢'(0) exists® and therefore lim (f) = @, = ¢ (0) exists, then J,— @,- £240 Proof The existence of lim @ (H)— (0) t>+0 ; implies the boundedness of this ratio in an interval of the form 0+0 of the function @(t) tend to ¢,. The above rules may at once be transferred to the Fourier series of an integrable function f(x), which we assume from the first to be given in 0 <2 < 2x and to be extended to all other real values of x by the equation 7 + 2 7) =7). In order that the Fourier series generated by f(x) should converge to a sum s(x,) at the point z,, the integrals 8 2 sin 2n+1)¢ I= 2g (2; @,) Sars: at 0 must, by theorem 9 (203), form a null sequence, where, as before, P(t 30) = 3 [F@ + 20) + F@, — 28] — 5 (x,). ~ This form of the criterion shows, over and above Riemann’'s theorem . 204, that neither the behaviour of f(x) immediately to the right of x, nor that immediately to the left of x,, have in themselves any influence ~ whatever on the behaviour of the Fourier series of f(x) at x,. What is important is that the behaviour of f(x) to the right of x, should stand in a certain relation to that on the left of x,, namely, such that the function (1) = @ (70) = 5 [f (@ + 28) + F(z, — 28] — 5 (ay) 7 8 Tt suffices that ¢’ (0) should exist as the right hand differential coefficient 9’ (4-0), as in fact the possible values of ¢ (£) for #<< 0 do not come into account. 370 : Chapter X1, Series ot variable terms. should possess the necessary and sufficient properties for the existence of the limit of Dirichlet’s integrals J, (208) relative to ¢ (2). It is not known what these properties are. The four conditions given above for the convergence of Dirichlet’s integrals furnish us, however, with the same number of sufficient conditions for the con- vergence, at a special point @,, of the Fourier series of a function f(x). Each of these conditions pu in the first instance, that the function ¢(t) =z) = lr +20) + fx, — 28) — 5 (2) should tend to a limit ¢,. A common assumption for all the rules which we are about to set up is accordingly the following: The limit (8) Jim 5 [fz +20) — Fm, — 29) must exist. The value of this limit, by theorem 2, will then also be the sum of the Fourier series of f(x) at x,, if the latter converges. This convergence is ensured if the function (0) = (5 20) = 5 [F®0 + 20) + Figg — 20] — s (zp), considered as a function of #, fulfils one of the four conditions given above. At the same time, the value ¢, in those conditions must, by theorem 2a, be 0. We accordingly assume that the two following conditions are satisfied: 207. 15t assumption. The function f(x) is defined and integrable (hence bounded) in the interval 0 < « < 2m and its definition is extended to all “veal values of x by means of the relation f(@) = fx +2ka), Lew), 4.8, 21d assumption. Ihe limit lim = fot 20 +f, — 21), t>+02 where , denotes an arbitrary real number, but is kept fixed throughout, exists, ond its value is denoted by s(x,), so that the function ® () = p(t 2) = g [Feo +20) + Fz — 24] — s(x) has a right hand limit lim @ (1) = 0. t>+0 With these joint assumptions, we have the following four criteria for the convergence of the Fourier series of f(x) at the point z,: 55 Define e. g. f(x) as entirely arbitrary to the right of z, (but integrable in an interval of the Jorm sm, ". 2. Dini’s rule. If for a fixed (otherwise arbitrary) positive num- ber O the integrals 8 { 2 dt : remain less than a fixed number for every © such that 0 < © <6, the Fourier series of f(x) converges at x, and its sum is s(x). - 8. Lipschitz’s rule. The same is true, if instead of requiring that the integrals should be bounded, we stipulate that two positive numbers A and o should exist, such that, for every t such that 0 +0 t->+0 should exist, and that the two functions @, (6) = (5g + 28) — Fw, +0) and gq, ()) = Fle, — 28) — f(z, — 0) should each, individually, satisfy the conditions of one of the four rules. ~ The Fourier series of f(x) is then convergent at z, and has the sum 1 s (¢y) = 5 [f(@, + 0) + f(x, — 0)]. One or two special cases, which, however, are of particular im- portance in applications, may be mentioned in the following further corollaries: 8 In case it converges at x,, the Fourier series of a function f(z) satis- fying the assumptions 207 accordingly has the sum f(z,) if, and only if, the limit s(z,), whose existence is stipulated in the second assumption, =f (x,). Similarly in the case of the following rules. 372 Chapter XI. Series of variable terms. Corollary 8. If f(z) satisfies the first assumption and is monotone both to the right and to the left of x, the limits mentioned in the preceding corollary exist, and the Fourier series of f(x) converges at x, to the sum s(x) = + [fy + 0) f(x, — 0)]. — Hence, still more particularly: - Corollary 4. The Fourier series of a function f(x) which satisfies the first assumption will converge at the point x, and its sum will be the value f(x) of the function at that point, if f(x) is continuous at x, and monotone on either side of z,. Corollary 8. If f(x) satisfies the first assumption, and the two limits f(x, 4 0) exist; if, further, both the (one-sided) limits qn LD SJE a f(x, +0) and lim f(x —h)— i —{) oy +0 h—>+0 exist; then the a? series of f(x) will converge at z, and will have the sum s(z,) = > rte, +0) + f(z, — 0)]. Corollary 6. The Fourier series of a function f(z) which satisfies the first assumption will converge, and will have as its sum the value of the function, at any point a, at which f(x) is differentiable. § 50. Applications of the theory of Fourier series. As we see from the rules of convergence developed above, extremely general classes of functions are represented by their Fourier series. This we propose to illustrate by a number of examples. The function f(x) to be expanded must always be given in the interval 0 < # < 2 # and must possess the period 2 a: f(z + 2 x) = f(x). The corresponding Fourier series is then, in general, obtained in the form 1 = : = % =e (a, cos nx +b, sinnz). In particular cases, the sine- or cosine-terms may be absent. In fact, if f(x) is an even function, f(—2)=fQa—2a)=/r(), (the graph of f(x) is symmetrical with respect to the straight lines g=hu, (=0,11, 1 2,..), and therefore 2x 7 2x 7b, = [f@)sinnede=[+ [=o as is evident if we spite x by — x in the second of these two partial integrals. The Fourier series of f(z) thus reduces to a pure bi oS asa § 50. Applications of the theory of Fourier series. 378 cosine-series. If, on the other hand, f (x) is an odd function, f(—d)=f@m—2)=—f(), (the graph of f(x) is symmetrical with respect to the points x=~kum, E=0, +1, + 2,...), and therefore 2x nea, = J f(@)cosnade = 0, 0 as is equally evident. Thus here the Fourier series of f(x) reduces to a pure sine-series. There are accordingly three different ways in which an arbitrary given function F(z), which is defined and integrable in a 2x, a portion of length 2x is cut out of the interval (a, b), say ¢« +0 exists, it will be advisable to modify our functions further at the 57 If b—a> 2x, a portion of the curve y= F(z) is left out of the re- presentation altogether, If we wish to avoid this, we need only alter the unit of measurement on the z-axis so that the interval of definition of F(z) has b—a 2x % Or else give the interval of definition of F(x) the exact length 2a by modifying the unit of measurement on the z-axis. the length 2x; i.e. we substitute x for z. 13% 5 S74 Chapter XI. Series of variable terms. - junctions 2 ks by writing (0) =@ka) = lim Lir@ + (27 — 2)] whenever this limit exists. (This is certainly the case for f(z), and provides the new condition f, (0) ={f,(2ks)=0.) If this limit does not exist, the functional value f(2kx) does not come into account, as with our resources we cannot discover whether the Fourier series converges there or not. — For corresponding reasons we have already put f; (wr) = 0 above. We now go on to concrete examples. 209. 1. Example. Fln)=a+=0. Here fi (x) =f, (x) = a, while we have to put @ Jor z—0mda=—umn fy (x) = a nw 0 sina 4]; p) for o = + p = integer 4 1 . 3 Ni 7 net sine 1, if p is even, Cos px = sin 2 x 4 tsinda fon], if p is odd. 4 27 2 22 — p? 42 p? 214. 215. TE 3 378 Chapter XI. Series of variable terms. From all the above series, innumerable numerical series may be deduced by taking particular values of x and «. 4. The treatment of F (a) =sina x leads to quite similar ex- pansions. 5. If the function F(z) = — log (2 sin 5 is arranged for the generation of a pure cosine series, we obtain the expansion, valid in QO 2< am, cos a + —; prim 3% + cos =— log 2 sin). It has, however, to : shewn by a special investigation that the result holds in spite of the fact that the function is unbounded in the neigh- bourhood of the points 0 and 27, and therefore is not (properly) inte- grable. (Cf. § 55, V below, where this will follow quite simply in another way.) 6. Example. F(z)=1¢%%-}¢7%% «= 0, is to be expanded in a cosine series. We have therefore to take f. @) ZH) ir 0< en Ty 2 Fi2n—%) In agen. After working out the extremely easy integrals giving the coefficients a, we obtain T e+e” ax aT ra —— Eset a TT secos2ax—+4---, which is valid in — an 1 +57), n=1 which accordingly is the Fourier series of its sum (v. 200, 1a) but which represents a function which is continuous but nowhere differen. tiable. 3. Whether continuous functions exist whose Fourier series are everywhere divergent is not at present known. “RY Q Ss SN Fig. 9. 4. A specially remarkable phenomenon is that known as Gibbs’ phenomenon 4, which was first discovered (by J. W. Gibbs) in connec- 2 We now have simpler examples than that mentioned above. E.g. L. Fejér has given a very clear and beautiful example (J. f. d. reine u. angew. Math., Vol. 137, p. 1.. 1909). 8 Abhandlungen zur Funktionenlehre, Werke, Vol. 2, p. 223. (First published 1875.) ” 64 J. W. Gibbs, Nature, Vol. 59 (London 1898-99) p. 606. — Cf. also T. H. Gronwall, Uber die Gibbssche Erscheinung, Math. Annalen, Vol. 72, p. 228, 1912 380 ~ Chapter XI. Series of variable terms. tion with the series 210Qa: The curves of approximation y=s, () overshoot the mark, so to speak, in the neighbourhood of x =0. More precisely, let us denote by & the abscissa of the greatest maxi- mum of y =s, (x) between 0 and z% and let 7, be the corresponding ordinate. Then & — 0; but 2, does not — 5, as we should expect, but tends to a value g equal to 5 (1089490 ...). Thus it appears that the limiting configuration to which the curves y=s (x) approxi mate, contains, besides the graph of the function 10a (p. 351, fig. 7), a stretch of the y-axis, between the ordinates = g, whose length ex- ceeds the “jump” of the function by nearly 9%. In fig. 9, the nth ap- proximation curve is drawn for » = 9 in the interval 0... sz, and for - n= 44 the initial portion is given. § 51. Products with variable terms. Given a product of the form 0 n=1 whose terms are functions of z, we shall define (in complete analogy with the theory of series) as an inferval of convergence of the product, an interval J at every point of which all the functions f, (x) are de- fined and the product itself is convergent. Thus e. g. the products 0 x2 © x2 © : x © x J(-5), J(1+5). J (1+6D 5): J (1+ 500m) are convergent for every real z, and the same is true of any product of the form II (1+ a,x), if 2a, is either absolutely convergent (v. 127, theorem 7) or a conditionally convergent series for which X'a,* converges absolutely (127, theorem 9). For every « in J, the product then has a specific value and there- fore defines a determinate function F(x) in J. We again say: the product represents the function F(x) in J, or: F(x) is expanded in the given product in J. The main question is as before: how far do the funda- mental properties (of continuity, differentiability, etc.) belonging to the terms f, (x), still hold for the function F(x) represented by the product? Here again the answer will be that this is the case in the widest measure, as long as the products considered are uniformly convergent. x 3x 5x nll ntl v1"? 272 4x first being the greatest maximum. The minima occur at z= Eire evi 65 The maxima in the interval occur at # = the § 51. Products with variable terms. ’ 381 What the definition of uniform convergence of a product is to be is almost obvious if we refer to the corresponding definition for series, since in either case we are essentially concerned with sequences of functions (cf. 190, 4). However, we shall set down the definition corresponding to the 27d form (191, 4) for series: © Definition ®¢. The product II(1 + f, (x) is said to be uniformiy 217. convergent in an interval J, if, given ¢ > 0, a single number N, (therefore independent of x), can be chosen so that |(1 + for: (@) (1 + Foi 2) bi (1 = fos1 @)) Fe 1] N, every k => 1 and every = in J%. It is not difficult to show that with this definition as basis the theorems of § 47 hold equally, mutatis mutandis, for infinite products ®s. We will, however, leave the details to the student, while we prove a few theorems which are less far-reaching, but which will amply suffice for all our applications, and which have the advantage of providing us at the same time with criteria for the uniformity of the convergence of a product. We first have OTheorem 1. The product II(1 + f, (x) converges uniformly in J 218. and represents a continuous function in that interval, if the functions f, (x) are all continuous in J and the series Xf, (x)| converges uni- formly in J. Proof. If 3'|f,(x)| converges in J, so does the product (1 +£, (2), by 127, theorem 7; indeed, it converges absolutely. Let F (x) denote the function it represents. Let us choose m so large that [ns d5 = (Tn 2) dere han) 1; this is possible, by hypothesis. Consider the product # I a+£®), n=m+1 % The symbol o in this section again holds only with, the same restric- tions as in §§ 46—48. 87 This definition includes that of convergence. If the latter be assumed, we may speak of the “remainder” 7, (z)= (1+ fp+1 @) (1 + fa+2@) ... and define uniform convergence as follows: II (1+ f, @) is said to converge uni- formly in J, if for every (x,) in J, however chosen, 7, (x) — 1. a. mn A 0 88 Writing 1a +f @) =P, (2) and JJ (A+4F, (x)) = F,, (x), we may de- y= r=m+1 duce e. g. the continuity of F(x) at x, quite easily from the relation F(x) — F (x) = Py, (2) F,, (x) — P,, (@o) + Fn (%,) =n (@) = Pp (%)] Fp () + [Fp (@) — Fo (2)] Pry (25) - v / . 3892 ; ~ Chapter XI. Series of variable terms. and denote its partial products by p (x), n >m. Let F, (x) be the function represented by this product. We have (cf. 180, 4) E,; == ; + (Psa To Das) 2 + (Pn Due) = it = Puts + Pmir’ its Pah ns trl sp fees, ie. F, (x) is also expressible by an infinite series — as is indeed evident from § 30. Now this series converges uniformly in J. In fact, for every n > m, we have | P| Sta) AE Dee AFL] < el Tm FH melt coc 3, iis 3 |£.@] n=m+1 is uniformly convergent in J, by hypothesis. Accordingly, (by 192, theorem 3) the Segueitts of functions p, (x) tends uniformly to F, in J, so that the product ir (1 +f, (x) is seen to converge uniformly in J, - Tt and this property is a E entid when we prefix the first m factors. By 193, F, (x) is necessarily continuous in J, since the terms of the series which represents it are all continuous in that interval. The same is then true of the function 3 F)=Q1+f@®)..- A+ 1, @)F,) ged A similar proof holds for _OTheorem 2. If the functions f, (x) are all differentiable in J and if not only X|f, (x)|, but Z| f @)] converges uniformly in J, then F(x) is also differentiable in J. Moreover its differential coefficient at every point of J where F(x) == 0 is given by F(x) — 2 fa (x) 69 Fe) S146 ‘Proof. The proof may be put in a form analogous to that of the previous theorem; however, in order to make other methods of attack familiar, we will conduct the proof by means of the logarithmic function, as follows. Let us choose m so large that Vora Io ia Go 6 If g(a) is differentiable at a special point z and g(z) 3 0 there, the ratio eo is called the logarithmic fierential coefficient of g(x), because it is 2 tog |g(@)|- For g(x)=g,() 8 (@) ... gk (x), we have, as is well known, £@) 60 &@, o/@ g(x) & (x) 8 (a) a @) provided that the functions g, () are all differentiable at the point z in question. \ . § 51. Products with variable terms. 383 for every x in J, so that, in particular, for every un > m, 1 | t: (x) | < 9 2 By 127, theorem 8, the series 3 log (1 +f, @) n=m+1 is then absolutely convergent in J. The series obtained from it by differentiating term by term, g I (x) wo TH ha @ is indeed also uniformly (and absolutely) convergent in J. For since If, @)] a and therefore EE < 2, so that the uniform convergence of the last series follows from that of X|f,’(2)|. Accordingly (by 196) Bf EE FE, €)) n=m+1 1 + Fr (2) if, as before, we put ALE) =F — Tg (+47, @) = log F, (x). Since finally er and the last factor on the right has been seen to be differentiable in J, F(z) itself is differentiable in J. If, further, F (x) == 0, the last relation leads at once to the required result, by the rule of differentiation men- tioned in the preceding footnote. Applications. ol. The product 219. 0 22 B(x) = = JI (1 -Z) is uniformly convergent in every bounded interval, since : 1 Sime aN is evidently a uniformly convergent series in that interval. The product ac- cordingly defines a function F (x) continuous everywhere. This function is also x2 n2 differentiable, for Jf (@)|=2]s]| =~ = is also uniformly convergent in every bounded interval. Hence for every xz se), 1 1.9... ; F@ 1 2 ow Ee et: n=1 884 Chapter XI. Series of variable terms. By 117, this however implies _(singa)y F (2) sinzz ’ Now if a relation holds between F (x) and F, (x) of the form El PEF) FF Breer it follows that d /F, =(2)-0 or Fy=C-F. Taking F, = sin wz, this gives © 22 singz=c-z- J] (1-5). n=1 where ¢ is a suitable constant. To determine its value, we need only divide the last relation by z and let z— 0. The left hand side then — x, as is shown by the power-series expansion of the sine function, while the right hand side —>c, because the product is continuous at x=0. Accordingly c=x and we have 2 : - x” sinte=nxx. [| t=). n=1 n 2. For cosz x we now find, without further calculation, COS TX = 2 2 JI (1 - nd sin2a2 ” nj (1 42? ) page Zen ff{1~2) Pek BET 8. The sine-product for special values of x leads to important numerical 1 product expansions. E. g. for B=» R=) FER). n=1 As Sat, we may clearly omit the brackets, and we accordingly write = 2-2-4-4.6-6-8.8... 2 1-3-8-5-5.7.7-9... (Wallis’ Product)™. Since it follows from this that 7 BY-8Y Gi oid ioe) \B 2k-1/ '2F+1 2 or 2k 1 — 246 / i ‘e BIE ETE “0 This product, and those discussed immediately below in 2. and 4., besides the remarkable product 287 and many other fundamental product-expansions, are due to Euler. M Arithmetica infinitorum, Oxford 1656. (Cf. pp. 218—9, footnote 1.) Exercises on Chapter XI. 385 we obtain at the same time the remarkable asymptotic relation ; 1 LCT) 928% n/" Van for the ratio of the middle coefficient in the binomial expansion of the (2x) power to the sum 22” of all the coefficients of this expansion, or for the co- efficient of x” in the expansion of i—= O04. Th. sequence of functions n!n?® n n (v. 128, 4) caanot be immediately replaced by a product of the form I7(14-f, (x), as (1 +) diverges for x 4 0. However, this divergence. is of such a kind that 1 2 1 2 1 2 2 (1454+) (1+ D143) (143) ~e By 128,2 and 42, 3, this implies that o(1+7) (1+) A. (1+) nx = gn (2) tends, as #»— 00, to a specific limit, finite and = 0; — the latter, of course, only if x + 0,—1, —2,.... Accordingly iol is a definite number for every #4 0, —1, — 2, .... The function of z so de- fined is called the Gamma-function (I'-function). It was introduced into analysis by Euler (see above) and, next to the elementary functions, is one of the most important in analysis. Further investigation of its properties lies out- side the scope of this book. (Cf., however, pp. 439—440.) Exercises on Chapter XI. I. Arbitrary series of variable terms. 154. Let (nx) denote the difference between nx and the integer nearest Ty : : ; : to z, or the value +5) if nx lies exactly in the middle of the interval between (nx) 72 two consecutive integers. The series 3) is uniformly convergent for all 2's. The function represented by it, however, is discontinuous for z= —2 — p ) 5 (p, q integers), while it is continuous for all other rational values of x and for all irrational values of z. : 155. If a,— 0, sin ney : 3a nw converges uniformly for all x's. Does this remain true for a, =1? 386 Exercises on Chapter XI. 156. The products : a) (1+ 2), b) Hen’, JI (1+ sine 2) 5 dy Jf (v FE 1" sin) converge uniformly in every bounded interval. = xh 157. The series whose partial sums have the values s, =r oT Zz? 1 converges for every z. Is this convergence uniform in every interval? Draw the curves of approximation. 158. A serics Xf, (x) of continuous positive functions certainly converges uniformly if it pk a continuous function F(x). (cf. p. 344.) 159. Does >} Sonyerge uniformly in every interval? Is the 7 (1 0 Zo function it represents continuous? 160. In the proof of 111, a situation of the following kind occurred: An expression of the form F (1) = a, (n) +a, (n) +++ ay (0) ++ + + ap, (3) is considered, in which, for every fixed k, the term a; (n) tends to a limit eo as n increases. At the same time, the number of terms increases, p,— 00. May we infer that led lim Fn) = Jwe;, n> k=0 provided the series on the right converges? Show that this is certainly per- missible if, for every % and every #, | ax (n) | remains tog 2 for z=+41. 8 What is their behaviour when z—1—0? — Examine the two series, con- vergent for x > 1, 1 2 1 2 le a ll vee 1 1 1 1 1 ee en Voi but 1. 3 ov "pw + 7 = for x—>-4+1-10. 2n—1 on bE 162. The series 5 Rr converges in 0 14-0, - 0 1 n=1 ; ZT 1 b) lim Py ——i—.|=C «176, 1). 164. Show that, for x —>1—0, = (— =a xn 1 a) = 7 Yar loge, b) 1-2-3 Prete lo 5 log2, iy 1 c) 1—2)- 2-1 Sa 165. The series whose partial sums have the values s, (x)= Tae may not be integrated term by term over an interval with endpoint 0. Draw the curves of approximation. II. Fourier series. 166. May we deduce from the series 210a, by integration term by term: & 1 2) > Se 22 (a mot) at 00 nay 2 a 5 x : = (got =o +a) az, y § sntnns ¢ cosdens i v2 Yaz} 4 y REL ew airal oo In EL intervals are these relations valid? 167. In the same way, deduce from 210 c the relations sin@n—N)z ax a) wp a Ht = 5 (1-2), Erie ) a ar b) 2 Bas slg (2+ 2m — 22%). What would be the results of further integrations? In which intervals are these expansions valid? 168. From 209, 210, and the relations in the two preceding exercises, deduce the following further expansions and determine their exact intervals of validity: cos3x cosh JT a) LOS Perrrmreh 5 eli Shae cling cos8xz , cosbz 7 (= b) OSE ~gp rrr Ey a), : sin» | sin} 2 2 ¢) sing — fr —e = (=), ete. 169. From 215, deduce further expansions by substituting x — 2 for = or by differentiating term by term. Is the latter operation allowed? What are the new series so obtained? 388 Chapter XII. Series of complex terms. 170. What are the sine-series and the cosine-series for ¢%®? What is the complete Fourier expansion of ¢S"%3 Show that the latter is of the form tly sing — a,cos2x—bysin3zx+a,cosdx+b;sindz——++--- where a, and b, are positive. 171. If x and y are positive and by <=, it appears that the modified conditions 4 are satisfied by another more general system of symbols, namely the system of ordinary complex numbers, but that no other system substantially different from the latter can satisfy them. 3. Accordingly, the system of (ordinary) complex numbers is a system of symbols — which, as is known, may be assumed to be of the form x 4-94, where « and y are real numbers, and { is a symbol whose manipulation is regulated by the simgle condition $2= —1, — for which the fundamental laws of arithmetic 2 remain valid without ex- ception, provided the symbols << and > are suitably replaced throughout by =. In short: Except for the lastnamed restriction, we may work formally with complex numbers exactly as with real numbers. 4. In a known manner (cf. pp. 7—8), complex numbers may be brought into (1, 1) correspondence with the points of a plane and may thus be represented by these: with the complex number z -- yi we associate the point (x, 9) of an zy-plane. Every calculation may then be interpreted geometrically. Instead of representing the number x 4 yi by the point (z, y), it is often more convenient to represent it by a directed line (vector) coincident in magnitude and direction with the line from (0, 0) to (z, y). 5. Complex numbers will be denoted in the sequel by a single ® letter: 2,7, 4,0,...; and unless the contrary is expressly mentioned : | ] or follows without ambiguity from the context, such letters will 7s- variably denote complex numbers. 390 Chapter XII. Series of complex terms. s 6. By the absolute value (or modulus) |z| of the complex number x-+yi, is meant the non-negative real value Va? 4 92; by its amplitude (amz, z= 0), we mean the angle ¢ for which both cos¢ = and sin p = =. When we calculate with absolute values, the rules 3, II, 1—4 hold unchanged, while 5. loses all meaning. - Since we may accordingly operate, broadly speaking, in precisely the same ways with complex as with real numbers, by far the greater part of our previous investigations may be carried out in an entirely analogous manner in the realm of complex numbers, or transferred to the latter, as the case may be. The only considerations which will have to be omitted or suitably modified are those in which the numbers themselves (not merely their absolute values) are connected by the symbol < or >. : In order to avoid repetitions, which this parallel course would otherwise involve, we have prefixed the sign © to all definitions and theorems, from Chapter II onwards, which remain valid word for word when arbitrary real numbers are replaced by complex numbers, (this validity extending equally to the proofs, with a few small alterations which will be explained immediately). We need only glance rapidly over the whole of our preceding developments and indicate at each place what modification is required when we transfer them to the realm of complex numbers. A few words will also be said on the subject of the somewhat different geometrical representation. Definition 23 remains unaltered. A sequence of numbers will now be represented by a sequence of points (each counted once or more than once) in the plane. If it is bounded (24, 1), none of its points lie outside a circle of (suitably chosen) radius K with origin at 0. Definition 285, that of a null sequence, and the theorems 26, 27, and 28 relating to such null sequences remain entirely unaltered. The sequences (z,) with wi oo fAVEY Lond (=r oi ¥ = (LEY, way wd 12,8000) #5 1 n are examples of null sequences whose terms are not all real. The student should form an exact idea of the position of the corresponding sets of points and prove that the sequences are actually null sequences. , The definitions in § 7 of roots, of powers in the general sense, and of logarithms were essentially based on the laws of order for real numbers. They cannot, therefore, be transferred to the realm of complex numbers in that form (cf. § 55 below). The fundamental notions of the convergence and divergence of a sequence of numbers (39 and 40,1) still remain unaltered, I as R § 52. Complex numbers and sequences. 391 . although the representation of z,—(' now becomes the following: If a circle of arbitrary (positive) radius & is described about the point { as centre, we can always assign a (positive) number », such that all terms of the sequence (z,) with index # > #, lie within the given circle. The remark 39,6 (1% half) therefore holds word for word, provided we interpret the e-neighbourhood of a complex number { as being the circle mentioned above. In setting up the definitions 40,2,3, the symbols << and > played an essential part; they cannot, therefore, be retained unaltered. And although it would not be difficult to transfer their main content to the complex realm, we will drop them entirely, and accordingly in the complex realm we shall call every non-convergent sequence divergent >. Theorems 41, 1 to 12, and the important group of theorems 4:3, with the exception of theorem 3, remain word for word the same, together with all the proofs. The most important of these theorems were the Cauchy-Toeplitz limit-theorems 43,4 and 5, and since we have in the meantime gained complete familiarity with infinite series, we shall formulate them once more in this place, with the extension indicated in 44, 10, and for complex numbers. Theorem 1. The coefficients of the matrix 221. Boor: Bogs Bons» oon gpnive os Hyg yyy gn hs oa lly 30a (A) oor Tay ogy w+ Tags Dror Ypis Yas +2 Tyr, vee are assumed to satisfy the two conditions: (a) the terms in each column form a null sequence, i. e. for every fixed n > 0, 4,0 as k—r00. 1 For complex numbers and sequences, we preferably use in the sequel the letters 2,(,Z,.... 2 We might say, in the case |z,|—>+ 00, that (z,) is definitely divergent with the limit 00, or tends or diverges (or even converges) to OO. That would be quite a consistent definition, such as is indeed constantly made in the theory of functions. However, it evidently involves a small inconsist- Ne ency relative to the use of the terms in the real domain, that e.g. the sequence of numbers (—1)"#% should be called definitely or indefinitely convergent, according as it is considered in the complex or in the real domain. And even though, with a little attention, this may not give us any trouble, we prefer to avoid the definition here. 222. 399 Chapter XII. Series of complex terms. (b) there exists a constant K such that the sum of the absolute values of any number of terms in any one row remains less than K, t. e., for every fixed k > 0, and any n: Jo El hela le RB. Under these conditions, when (24, z;,...) is any null sequence, the numbers z/ 207% = © % + 1 = 2a, Zn also form a mull sequence?®. Theorem 2. The coefficients a,, of the matrix (4), besides satis- fying the two conditions (a) and (b), are assumed to satisfy the further condition (c) A eet Pa as k—oo?d, n=0 In this case, if z,— (, we have also oo 5 =a, ta, t= = Tn z, —¢. (For applications of this theorem, see more especially 233, as well as §§ 60, 62 and 63.) Unfortunately, we lose the first of the two main criteria of § 9, which was the more useful of the two. Moreover, the proof of the second main criterion cannot be transferred to the case of complex numbers, as it makes use of theorems of order throughout. In spite of this, we shall at once see that the second main criterion itself — in all its forms — remains valid for complex numbers. The proof may be conducted in two different ways: either we reduce the new (complex) theorem to the old (real) one, or we construct fresh founda- tions for the proof of the new theorem, by extending the develop- ments of § 10 to complex numbers. Both ways are equally simple and may be indicated briefly: 8 1. The reduction of complex sequences to real sequences is most easily accomplished by splitting up the terms into their real and imaginary parts. If we write z, =, iy, and { =¢& | iy, we have the following theorem, which completely reduces the question of the convergence or divergence of complex sequences to the corresponding real problem: 1 Theorem 1. The sequence (z,) = (x, + 1y,) converges to { =& tn if, and only if, the real parts 2) converge to & and the imaginary parts ¥, converge to 1). ; 3 In consequence of (b), 4x = 2 ax, is absolutely convergent and there- 7 fore, as the z's are in themselves }onnded, the series 2 in z, =z; is also 1 absolutely convergent. -~ § 52. Complex numbers and sequences. 393 . Proof a) x —& and y,—, (zr, — &) and (y, — n) are null sequences. By 26,1, the same is true of i(y, — 5) and, by 28,1, of (=, he £) wf i (yy = 7) Le. of (2, > £). b) If z,—(, |z,— {| is a null sequence; since Jo alent and |= Clr Sal, (x, — &) and (y, — 7) are also null sequences, by 26,2, 1. e. we have both : z,—£ and Vos The theorem is established. The theorem at which we are aiming follows immediately: Theorem 2. For the convergence of a complex sequence (z,), the conditions of the second main criterion 4d are again necessary and sufficient, — namely, that, for every choice of ¢ > 0, we should be able to assign mn, so that [Zw — 20] < es, for every n> mn, and every w' > n,. Proof. a) If (z,) converges, so do (x,) and (y,) by the preceding theorem. As these are real sequences, we may apply 47, and, given ¢ > 0, we may choose n, and n, so that [oy —, [<< - for every n >», and every #’ > u,, and | yw — 9, | < 5 for every n > n, and every #' > u,. Taking n, greater than n, and #,, we have accordingly, for every n > #n, and every #' > n,, : EM = = | (@n — 2 toy = Ya) | = | Tn z, | + | yw = Yul & & : = ag -} 9 = &. The conditions of our theorem are therefore necessary. b) If, conversely, (z,) fulfils the conditions of the theorem, — i e. given ¢ > 0, we can determine n, so that | z,» — z, | < & provided only that # and »’ are both > ny, — we have also, for the same # and #»’ (by our last footnote) | —z,| 0, the relation lz, —C| any given n,)- Theorem. Every bounded sequence possesses at least ome limiting point. (Bolzano- Weierstrass Theorem.) Proof. Suppose |z,| < K and draw the square Q, whose sides lie on the parallels to the axes through +4 K and + 7K. All the zs 5 This statement at the same time expresses, in pure arithmetical lang- uage, the relations of magnitude framed in geometrical form in the theorem and definition 223 § 52. Complex numbers and sequences. 395 are contained in it, i. e. certainly an infinity of z's. Q, is divided by the coordinate-axes into four equal squares. One at least of the four must contain an infinity of z's. (In fact, if there were only a finite number in each, there would also be only a finite number in Q,, which is not the case). Let OQ, denote the first quarter” which has this property. This we again proceed to divide into four equal squares, denoting by Q, the first quarter which contains an infinity of points z,, and so on. The sequence Q,, Q,, OQ,, ..- forms a nest of squares, since each (, lies within the preceding and the lengths of the sides form a null sequence, namely (2 Kg). Let ¢ denote the innermost point of this nest?; { is a point of accumulation of (z,). For if ¢ is given > 0 and m is chosen so that the side of (, is less than os of {, and, with it, an infinite number of points z, also lie in this neighbourhood. Therefore { is a point of accumulation of (z,), and the existence of such a point is established. the whole of the square (Q lies within the e-neighbourhood The validity of the second main criterion for the complex domain, — i.e. of the theorem 222,2, formulated above — may now be established once more, but without any appeal to the “real” theorems, - on the same lines as in 47. Proof. a) ff z,—¢, i.e. (2, — 7) is a null sequence, we can determine #7, so that 2, — Cl <5 and |m— Cl <5 provided only that # and #’' are simultaneously > n, [see part a) of the proof of 47]. For these n’s and #”s, we therefore also have 2 — ow | S| — C+ |2 = E] n, and n > m, J, =. I m lies in the circle of radius & round Zand Waking K to be larger than all the m numbers |z, |, lz], «-, 1%2,-1]> |2.| + & we have |z | < K for every n. 8 We regard the four quarters as numbered in the order in which the ~ four quadrants of the zy-plane are habitually taken. 3 ? The process of obtaining this point corresponds exactly to the method of successive bisection so often applied in the real domain. 226. 396 Chapter XII. Series of complex terms. By our preceding theorem, it follows that (z,) has at least one limiting point {. Supposing there exists a second limiting point {’' +, choose shine pf, which is positive. By 224, the definition of limiting point, we can choose n, as large as we please and yet have an n > n, for which |z,— {| mn, for which |z, — {| &® This contradicts our hypothesis. Accordingly { must be the unique limiting point, and outside the circle of radius e round { there is only a finite number of points z,. If n, is suitably chosen, we there- fore have |z, — (| m,, and consequently z,—¢. The condition of the theorem is therefore sufficient also °. § 63. Series of complex terms. As a series 2g of complex terms must obviously be interpreted as the sequence of its partial sums, the basis for the extension of our theory of infinite series has already been provided by the above. Corresponding to 22,1, we have first the Theorem. A series 2 a, of complex terms is convergent if, and only if, the series 2 R(a,) of the real parts of its terms and the series 2 3(a,) of their imaginary parts converge separately. Further, if these two series have the sums s' and s” respectively, the sum of 2X a, is =x -1-4¢", In accordance with 222, 2 the second principal criterion (81) for ‘the convergence of infinite series remains unaltered in all its forms, and, at the same time, the theorems 83 deduced from it, on the algebra of convergent series, also retain their full validity. Since, in the same way, theorem 85 also remains unchanged, we shall, as before, distinguish between absolute and non-absolute con- vergence of series of complex terms (Def. 86). ; 9 Zt ==" =O) +E =) + —2), hence lt —2 | Z| = C= |ow = |= | —C[>Bs—s—s=2. 9 Hence we may also say: (z,) converges if, and only if, it is bounded : and possesses only one point of accumulation. This is then at the same time the limit of the sequence. § 53. Series of complex terms. 397 Here again we have the Theorem. The series 2 a, of complex terms is absolutely con- 227. vergent if, and only if, both the series Z 5.) and 2 J (a,) are ab- solutely convergent. The proof results simply from the fact that every complex number z =x + 1 y satisfies the inequalities 2s In consequence of this simple theorem, it is at once clear that, with series of complex terms as with real series, the order of the terms is immaterial if the series converges absolutely (Theorem 88,1). If, however, 2g is not absolutely convergent, either Z'% (a,) or 2 3 (a,) must be conditionally convergent. By a suitable rearrangement of the terms, the convergence of the series 2a, may therefore be des- troyed in any case, as in the proof of theorem 89, 2, that is: In the case of series of complex terms also, the convergence, when it is not absolute, depends essentially on the order of succession of the terms. (Regarding the extension to series of complex terms of Riemann’s rearrangement theorem § 44, cf. the remarks on the following page.) The next theorems, 89, 3 and 4, as also the main rearrangement theorem 90, which relate to absolutely convergent series, still remain valid, without modification or addition, for series of complex terms. Since the determination of the absolute convergence of a series is a question relating to series of positive terms, the whole theory of series of positive terms is again enlisted for the study of series of complex terms: Everything that was proved for absolutely convergent series of real terms may be utilized for absolutely convergent series of complex terms. If we omit power series from consideration for the present, we ~ observe, on looking over the later sections of Part II (§§ 18— 27), that the developments of Chapter X are the first for which there is any question of transference to series of complex terms. Abel's partial summation 182, being of a purely formal nature, and its corollary 183, of course hold also for complex numbers, and so does the convergencetest 184 which was based directly on them. The special forms of this test may also all be retained, provided we keep to the convention agreed on in 220, 5, in accordance with which all sequences assumed to be monotone are ipso facto real. In the case of du Bois-Reymond’s and Dedekind’s tests, even this precaution ~ becomes unnecessary: they hold word for word and without any restriction for arbitrary series of the form Xa, b,, with complex a, and b,. Riemann’s rearrangement theorem (§ 44) is, on the contrary, essen- 14 228. 398 Chapter XII, Series of complex terms, tially a “real” theorem. In fact, if a series 2'a, of complex terms is not absolutely convergent, so is one at least of the two series 2'9i(a,) and 23 (a,), by 227. By a suitable rearrangement, we can therefore, in accordance with Riemann’s theorem, produce in one of these two series a prescribed type of convergence or divergence. But the other one of the two series will be rearranged in precisely the same manner; and there is no immediate means of foreseeing what the effect of the rearrangement on this series or on 2'q, itself will be. — It has recently been shown, however, that if 2'g_ is not absolutely convergent, it may be transformed by a suitable rearrangement into a series, again con- vergent, whose sum may be prescribed to have either any value in the whole complex plane or any value on a particular straight line in this plane, according to the circumstances of the case’. The theorems 188 and 189 of Mertens and Abel on multi- plication of series (§ 45) again remain valid word for word, together with the proofs. For the second of these theorems we must, it is true, rely on the second proof (Cesaro’s) alone, as we have provisionally skipped the consideration of power series (cf. later 232). At this point we are in possession of the whole machinery required for the mastery of series of complex terms and we can at once proceed to the most important of its applications. Before doing so, however, we shall first deduce the following ‘extremely far-reaching criterion. © Weierstrass’ criterion. A series > a, of complex terms, for which n=0 agri op as dp n nh with A, bounded, — where « is complex and arbitrary, and 2 >1 12, — 10 We thus have the following very elegant theorem, which in a certain sense completes the solution of the rearrangement problem: The “range of summation” of a series Xa, of complex terms — i. e. the set of values which may be obtained as sums of convergent rearrangements of Xa, — is either a definite point, or a definite straight line, or the entire plane. Other cases cannot occur. A proof is given by P. Lévy (Nouv. Annales (4), Vol. 5, p. 506. 1905), but an unexceptionable statement of the proof is not found earlier than in E. Steinitz (Bedingt konvergente Reihen und konvexe Systeme, J. f. d. reine u. angew. Math., Vol. 143, 1913; Vol. 144, 1914; Vol. 146, 1915.) 1 J. f. d. reine u. angew. Math, Vol. 51, p. 29, 1856; Werke I, p. 185. 12 An equality of this kind may of course always be assumed; we need i Z. a 7 hoh ; iy only write 4, = nk (x — i= th as a definition. What is essential in the n condition is here, as previously (cf. footnote to 166), that when « and 1 are suitably chosen the 4,’s should be bounded. — It is substantially the same thing to assume that ay, a B, =14+—4— Gn +1 t+ #7 ¥ nt with 4>1 and B, bounded. § 53. Series of complex terms. : 399 is absolutely convergent if, and only if, R(¢) >1. For R(«) <0 the series 1s invariably divergent. If 0 < R(«) <1, both the series RD eel = | (a, aa A) | and ol y Da n= = are convergent. Proof. 1. Let ¢=/-} iy and let us first-assume f= R(e) >1. In that case, if 4 < K, say, we write, as is permissible, an Eis Bgl AE and it follows at once that, if p’ is any number such that 1 < <8, Ap +1 = {i= B ' ies n a, for every sufficiently large n. By Raabe’s test, the series Xa, | is therefore convergent. 2. Now suppose R(¢)= <1. In that case, since El An +1 an for sufficiently large values of #, it follows from Gauss’s test 172 that 2|a,| is divergent. 3a. If, moreover, ft («) = f < 0, our last inequality shows that then it Therefore 2a, must now diverge. 3b. ¥ R(e)=4=0, i.e Buty ly ap a, n nk it is easy to verify that we then have An PT An +1 an =1— where A’ > 1and is the smaller of the two numbers 2 and 4, and the 4, ’s are again bounded. Accordingly, if ¢ denotes a suitable constant, An +1 a, >1—-5->0 13 As regards the series 2 a, itself, it was shown by A. Pringsheim (Archiv ~ d. Math. und Phys. (3), Vol. 4, pp. 1—19, in particular pp. 13—17. 1902) that this is also divergent whenever 9 («) < 1. The proof is a little more trouble- some and the fact itself is not used in our treatment. — A further more exact investigation of the series Xa, itself in the case 0< R(«) <1 is given by J. A. Gmeiner, Monatshefte- f. Math. u. Phys., Vol. 19, pp. 149—163. 1908. 400 Chapter XII. Series of complex terms. for every n > m, say. It follows by multiplication that 7 — 5) > 1 (1 —%)=C,>0. An Am Am +1 ap I Hence |a,|>C,,-|a, |, for every n >m, and a, cannot tend to 0, so that 2g again diverges (cf. 170, 1). 4. If, finally, %(¢)=p > 0, we have to show that both the series 2la,—a and X(—1)"a, n+1 are convergent. Now as in 1. we have, for every sufficiently large #, B’ . ’ = on with Bc f> 0. Accordingly, a n a) the series 2 (|a,| —|a,,,|) is convergent, by 131, and has, moreover, all its terms positive for sufficiently large #n’s. Now 1 LL Horr x ay, — ay +1 | to Age lan | —|@n11| 1 or) = 7 3 n since the fraction on the right hand side tends to the positive limit 2 when #— --o0, that on the left is, for every sufficiently large #, less than a suitable constant 4. By 70, 2, this means that 3|a, —a, | converges with (| a, | — | a,.,|). — We can show more precisely, how- ever, that b) a,— 0. For it again follows, by multiplication, from ’ that ; Srl Se (10555) The right hand side (by 126, 2) tends to 0 as n— -- co, hence (cf. 170, 1) we must have a,— 0. Now the series (2 — 2) + (0s — 23) + (0, — a) tn = 3 (a, — Gyr t1) is a sub-series of X(a, —a,,,) and therefore converges absolutely, by a); also, since |a,|+|a,.,|—0 with a, we may omit the brackets, by 83, supplement to theorem 2. This proves the con- vergence of 2 (—1)"a,. : This theorem enables us to deduce easily the following further theorem, which will be of use to us shortly: ; § 54. Power series. Analytic functions. 401 Theorem. If, as in the preceding theorem, Bott ot nl A, 4 arbitrary, 4 >1, He ew (4,) bounded, the series 2a, 2" is absolutely convergent for |z| < 1, divergent for every |z| > 1, and for the points of the circumference |z|=1, the series will a) converge absolutely, if R(e) > 1, b) converge conditionally, if 0 < R(e) <1, except possibly for the single point z= 114, c) diverge, if R(x) £0. Proof. Since Va, vg 2232 Ay 2 = |=}; the statements relative to |z|=1 are immediately verified. For |z| =1, the statement a) is an immediate consequence of the con- vergence of X|a, | ensured by the preceding theorem. Similarly c) is an immediate consequence of the fact established above, that in this case |a,| remains greater than a certain positive number for every sufficiently large #. Finally, if 0 < %(¢) <1 and z 41, the convergence of Xa z" follows from Dedekind's test 184, 3. For we proved in the preceding theorem that 2'|a,— a,,,| converges and g,— 0; that the partial sums of Xz" are bounded, for every (fixed) 2 <4=-1 on the circum- ference |z|=1, follows simply from the fact that for every = (1 —gntti [1+z4224-- 42" |= |——— 1—2z 2 =T1—7] § 54. Power series. Analytic functions. The term “power series” is again used here to denote a series of the form Za, 2", or, more generally, of the form Sa, (z — z,)", ~ where now both the coefficients 4, and the quantities z and z, may = be complex. The theory of these series developed in §§ 18 to 21 remains valid without any essential modification. In transferring the considerations of those sections, we may therefore be quite brief. Since the theorems 93,1 and 2 remain entirely unaltered in the new domain, the same is true of the fundamental theorem 93 itself, on the behaviour of power series in the real domain. Only the geo- metrical interpretation is somewhat different: The power series Ja, z" “If we take into account Pringsheim’s result mentioned in the preceding : footnote, we may state here, more definitely: except for z=+1. - 229. 230. 402 Chapter XII. Series of complex terms. converges — indeed absolutely — for every z interior to the circle of radius 7 round the origin 0, while it diverges for all points outside that circle. This circle is called the circle of convergence of the power series — and the name radius applied to the number 7 thus becomes, for the first ime, completely intelligible. Its magnitude is given as before by the Cauchy-Hadamard theorem 94. : : Regarding convergence on the circumference of the circle of con- vergence, we can no more give a general verdict than we could re- garding the behaviour at the endpoints of the interval of convergence in the case of real power series. (The examples which follow immedia- tely will show that this behaviour may be of the most diverse nature.) The remaining theorems of § 18 also retain their validity unaltered. Examples. 1. Iz; r=1. In the interior of the unit circle, the series is convergent, : 1 : th ; with the sum at On the boundary, i.e. for |z|=1, it is everywhere di- vergent, as z® does not — 0 there. zr . : : 2, Sr r=115, This series remains (absolutely) convergent at all the boundary points |z|=1. n 3. I=; r=1. The series is certainly not convergent for all the : ; ; : 1 i boundary points, for z=1 gives ihe divergent series —-,. However, it is also g g n ? not divergent for all these points, since z=—1 gives a convergent series. In fact, theorem 229 of the preceding section shows, more precisely, that the series must converge conditionally at all points of the circumference |z|=1 different from +1; for we have here a, s1—1 1 ee dn-3 n 7 The same result may also be deduced directly from Dirichlet's test 184, 2, since Xz" has bounded partial sums for z= +1 and |z|=1 (cf. the last formula of | the preceding section) and = tends monotonely to 018. As pits wl ’ n the convergence can, however, only be conditional. an 3 4. I r=1. This series diverges at the four boundary points 1-1 n ] and +1, and converges conditionally at every other point of the boundary. 3 ; 15 If Ya,z" has veal coefficients (as in most of the subsequent examples) this power series of course has the same radius as the real power series Xa, x". 18 These facts regarding convergence may also be deduced from 183, 5 by splitting up the series into its real and imaginary parts. Conversely, how- ever, the above mode of reasoning provides a new proof of the convergence of these two real series. 2 ria § 54. Power series. ' Analytic functions, 408 az» : 5. For rs r=-4+00. For Zn!z* r=0; thus this series converges nowhere but at z=0. 6. The series Se 1 ——— are everywhere 2k 2+ dN —_— oT Pe Pls i RT convergent. 7. A power series of the general form Xa, (2 — 2)" converges absolutely at all interior points of the circle of radius » round z,, and diverges outside this circle, where » denotes the radius of Xa, 2". Before proceeding to examine the properties of power series in more detail, we may insert one or two remarks on Functions of a complex variable. If to every point z within a circle & (or more generally, a domain? ¢) a value w is made to correspond in any particular manner, we say that a function w= f(z) of the complex variable z is given in this circle (or domain). The correspondence may be brought about in a great number of ways (cf. the corresponding remark on the concept of a real function, § 19, Def. 1); in all that follows, however, the func tional value will almost always be capable of expression by an explicit formula in terms of z, or else will be the sum of a convergent series whose terms are explicitly given. Numerous examples will occur very shortly; for the moment we may think of the value w, for instance, ~ which at each point z within the circle of convergence of a given power series represents the sum of the series at that point. The concepts of the limit, the continuity, and the differentiability of a function are those which chiefly interest us in this connection, and their definitions, in substance, follow precisely the same lines as in the real domain: 1. Definition of limit. If the function w= f(z) is defined for 237y, every z in a neighbourhood of the fixed point £18, we say that z>¢ or (=> w "for 27, 17 A strict definition of the word “domain” is not needed here. In the sequel, we shall always be concerned with the interior of plane areas bounded by a finite number of straight lines or arcs of circles, in particular with circles and half-planes. 18 f(2) need not be defined at the point itself, but only for all z's which satisfy the condition 0 << |2— {| <'¢. The § of the above definition must then of course be assumed 0, we can assign 6 =0 (¢) > 0 so that |7(0) — wo] w as z—{ along that arc, or within that angle, or in that set M, if the above conditions are fulfilled, at least for all points z of the set A/ which come into consideration in the process. 2. Definition of continuity. If the function w = f(z) is defined in a neighbourhood of { and at { itself, we say that f(z) is continuous at the point C, if lim f(z) z>¢ exists and is equal to the value of the function at {, i e. if f(z) — FO. We may also define the continuity of f(z) at { when z is restricted to an arc of a curve containing the point {, or an angle with its vertex at {, or any other set of points M of which { is a limiting point; the def initions are obvious from 1. - 3. Definition of differentiability. If the function w = f(z) is de- fined in a neighbourhood of { and at { itself, f(z) is said to be differ entiable at C, if the limit TR he £19] f z>¢ = exists in accordance with 1. Its value is called the differential coeffi- cient of f(z) at { and is denoted by f’({). (Here again the mode of variation of z may be subjected to restrictions.) We must be content with these few definitions concerning the general functions of a complex variable. The study of these functions in detail constitutes the object of the so-called theory of functions, one of the most extensive domains of modern mathematics, into which we of course cannot enter further in this place 2°. 19 Same proof as in the real domain. 20 A rapid view of the most important fundamental facts of the theory of functions may be obtained from two short tracts by the author: Funktionen- _ § 54. Power series. Analytic functions. T= 40% The above explanations are abundantly sufficient to enable us to transfer the most important of the developments of §§ 20 and 21 to power series with complex terms. : In fact, those developments remain valid without exception for our present case, if we suitably change the words “interval of conver- gence” to “circle of convergence” throughout. Theorem 5 (99) is the only one to which we can form no analogue, since the concept of integral has not been introduced for functions of a complex argument. All this is so simple that the reader will have no trouble, on looking ~ through these two sections again, fo interpret them as if they had been intended from the first to relate to power series with complex terms. At the most, a few remarks may be necessary in connection with Abel's limit theorem 100 and theorem 107 on the reversion of a power series. In the case of the latter, the convergence of the series yp, y> ++, and hence of the series y-}- 5,32 -}-..., which satis- fied the conditions of the theorem, were only proved for real values of y. This is clearly sufficient, however, as we have thereby proved that this power series has a positive radius of convergence, which is all that is required. As regards Abel's limit theorem, we may even — corresponding to the greater degree of freedom of the variable point 2 — prove more than before, and for this reason we will go into the matter once more: Let us suppose Xa z" to be a given power series, not everywhere convergent, but with a positive radius of convergence. We first observe that, exactly as before, we may assume this radius = 1 without intro- ducing any substantial restriction. On the circumference of the circle of convergence, |z|=1, we assume that at least one point Zz, exists at which the series continues to converge. Here again we may assume that z, is the special point 1. In fact, if 2,4 41, we need only put we by a, 3 = a, ’ the series 2a 2" also has the radius 1 and converges at the point -- 1. The proof originally given, where everything may now be interpreted as “complex”, then establishes the Theorem. If the power series 2 a, z" has the radius 1 and remains 232. convergent at the point =~ 1 of the unit circle, and if Za, =s, then we also have 2 n Im {Sy 2) =s z->+1 if z approaches the point 41 along the positive real axis from the origin 02%. theorie, I. Teil, Grundlagen der allgemeinen Theorie, 2°d ed., Leipzig 1926; II. Teil, Anwendungen und 'Weiterfithrung der allgemeinen Theorie, 20d ed., Leipzig 1926 (Sammlung Goschen, Nos. 668 and 703). 21 We are therefore dealing with a limit of the kind mentioned above in 281, 1. ree 233. 406 Chapter XII. Series of complex terms. We can now easily prove more than this: Extension of Abel’s theorem. With the conditions of the preceding theorem, the relation iIm(S a, P=1s z>+1 vemains true if the mode of approach of z to -+1 is restricted only by the condition that z should remain within the unit circle and in the angle between two arbi- trary (fixed) rays which pene- trate into the interior of the unit circle, starting from the point 4-1 (see Fig. 10). The proof will be con- ducted quite independently of previous considerations, so that we shall thus obtain a third proof of Abel's theorem. Let 25, 2,,4:+: 2,5 v0 the any sequence of points of Fig. 10. limit +1 in the described portion of the unit circle We have to show that f (7,)—>s if, as before, we write Xa 2" = f(z). In Toeplitz theorem 221, 2 choose for q,, the value Yin = ef ee 2) 2%, pe and apply the theorem to the sequence of partial sums Sp =a + ay +o Fa, which, by hypothesis, converges to s. It follows immediately that = =n) 55, =~) 2 54 == 5 =f) also tends to s as k increases. This proves the statement, provided we can show that the chosen numbers g,, satisfy the conditions (a), (b) and (c) of 221. Now (a) is clearly fulfilled, as z,—1, and the sum of the kth row is now A =0-u 2 ==1, so that (c) is fulfilled. Finally (b) requires the existence oF 2 constant K such that a 1-2-2] |= Edw for all points z=12z, <= --1 in the angle (or any sector-shaped portion of it with its vertex at --1). It only remains, therefore, to establish | ] § 54. Power series. Analytic functions. 407 the existence of such a constant. This reduces (v. Fig. 10) to proving the following statement: If z=1— go (cos ¢ +i sin @) with |p | < p, < % and 0 < 0 < 9, < 2cosp,, aconstant A = A(p,, 0,) exists, depending only on @, and o,, such that [1-2] = =4 for every z of the type described. In the proof of this statement, it is sufficient to assume g, = cos ¢,, and in that case we may at once 2 . 2 \ show that A YT is a constant of the desired kind. In fact, the 0 statement then runs: 0 2 1-J1—2gcosp +g = S959 or —2pcosp +0? < — cos p, + 4 0% cost gy, for 0 < pL cos, and |p| < ¢,. By replacing ¢ by ¢, and p? by ocos p, on the left hand side, the latter is increased; therefore it cer- tainly suffices to show that — cos, < — 0 cos @, + 7 0% cos? gp, — which is obviously true. — This extension of Abel's theorem to «complex modes of approach” or “approach within an angle” is due to 0. Stolz *, This completes the extension to the case of complex numbers of all the theorems of §§ 20 and 21 — with the single exception of the theorem on integration, which we have not defined in the present connection. In particular, it is thereby established that a power series in the interior of its circle of convergence defines a function of a complex variable, which is continuous and differentiable — the latter “term by term” and as often as we please — in that domain, and accordingly possesses the two properties which above all others are required, in the case of a function, for all purposes of practical application. For this reason, and on account of their great importance in further developments of the theory, a special name has been reserved 22 Zeitschrift f. Math, u. Phys., Vol. 20, p. 369, 1875. In recent years the question of the converse of Abel's theorem has been the object of numerous investigations, — i. e. the question, under what (minimum of) assumptions relat- ing to the coefficients a,, the existence of the limit of f(z) as z— 1 (within the angle) entails the convergence of Xa,. An exhaustive survey of the present state of research in this respect is given in a paper by G. H. Hardy and J. E. Littlewood, Abel's theorem and its converse, Proc. Lond. Math. Soc. (2), Vol. 18, pp. 205—235, 1920. — Cf. also theorems 278 and 287. 234. 235. 408 Chapter XII. Series ot complex terms. for functions representable in the neigbourhood of a point z, by a power series Xa _(z — z,)". They are said to be analytic or regular at z,. By 99, such a function is then analytic at every other interior point of the circle of convergence; it is therefore said simply to be analytic or regular in this circle®®. In particular, a series every- where convergent represents a function regular in the whole plane, which is therefore shortly called an integral function. All the theorems which we have proved about functions expressed by power series are theorems about analytic functions. Only the two following, which are of special importance in the sequel, need be ex- pressly formulated again. 1. If two functions are analytic in one and the same circle, then so are (by § 21), their sum, their difference, their product and their ratio; the latter, of course, only at points where the function in the denominator == 0. 2. If two functions, analytic in one and the same circle, coincide in a neighbourhood, however small, of its centre (or indeed at all points of a set having this centre- as point of accumulation), the two functions are completely identical in the circle (Identity theorem for power series 97). Besides stating these two theorems, which are new only in form, we shall prove the following important theorem, which gives us some information on the connection between the moduli of the coefficients of a power series and the modulus of the function it represents: Theorem. If f(z) = Na, (z — z,)" converges for |z — zy | <7, then n=0 1 M : EARS (p=0,1,2,..) if 0a, n since the ratio is equal to a for every #, in this case. If we consider the rather more general function gmp ld py be de 5, L0G nr) E—2)" I2% : vf by, (2 min YR, where ! and m are fixed integers > 0, and now form the arith- metic mean Sot Gtr duet n J (where, as before, g, = g(2, + 0%"), »=0,1,...), this clearly —b,, by the two cases just treated. If, further, it is known that the function g(z), for every z of the circumference |z — z,| = g, is never greater than a certain constant K, we have also n hist tel LMR _g, and therefore also 5] < K. With these preliminary remarks, the proof of the theorem is now ~ quite simple: Let p be a specific integer 20. As Ya lo” con * verges, given ¢ > 0, we can determine gq > p so that fa 01] + | agen] 0?" 24 - < ee. 410 Chapter XII. Series of complex terms. A fortiori, we then have for all values of z such that |z —z,| =p, | > a,(z — 2.7] The function between the modulus signs is of the kind just considered. The inequality |b,| < K there obtained now becomes Ca M [nls and, as ¢ was arbitrary and > 0, we have, in fact, (cf. footnote to 41, 1) = M a, | = nl q. 4d. § 55. The elementary analytic functions. I. Rational functions. 1. The rational function w = =. is expressible as a power series for every centre z, <4 1: 1 1 1 1 o 1 ; = w= 3 ee (7 — 2): 1-2 1—2z,— J i—7 i fd 2a ( 0) and this series converges for |z —z,| <|1—z,| i. e. for every z nearer to z, than 4-1; in other words, the circle of convergence of the series is the circle with centre z, passing through the point 4 1. The function = is thus analytic at every point different from 1. With reference to this example, we may briefly draw attention to the following phenomenon, which becomes of fundamental importance in the theory of functions: If the geometric series Xz”, whose circle of convergence is the unit circle, is expanded by Taylor's theorem about a new centre z, within the unit circle, we could assert with certainty, by that theorem, that the new series converges at least in the circle of centre z, which touches the unit circle on the inside. We now see that the circle of convergence of the new series may very possibly extend beyond the boundary of the old. This will always be the case, in fact, when 2, is not real and positive. If z, is real and negative, the 4 new circle will indeed include the old one entirely. (Cf. footnote to 99, p. 176.) § 55. The elementary analytic functions, — I. Rational functions. 411 2. Since a rational integral function a,+a,z+a, 2+ Fa, zm may be regarded as a power series, convergent everywhere, such functions are analytic in the whole plane. Hence the rational functions of general type ay+a,z4+---+a,z" by + by z+. by2* are analytic at all points of the plane at which the denominator is not 0, — i. e. everywhere, with the exception of a finite number of points. Their expansion in power series at a point z,, at which the denominator is = 0, is obtained as follows: If z is replaced by z, + (z — z,) both in the numerator and denominator of such a function, these being then rearranged in powers of (z — z,), the function takes the form z alt a 2 ental ry? by +b @—2) +--+ 0 (—7)* ’ where, on account ot our assumption, b,’ 40. We may now carry out the division in accordance with 103, 4 and expand the quotient in the required power series of the form Jc, (z — z,)" 2°. II. The exponential function. The series z 22 az 1th tm Fa is a power series converging everywhere, and therefore defines a func- tion regular in the whole plane, i. e. an integral function. To every point z of the complex plane there corresponds a definite number wu, the sum of the above series. This function, which for real values of z has the value ¢? as de- fined in 38, may be used to define powers of the base ¢ (and then further those of any positive base) for all complex exponents: 26 An alternative method consists in first splitting up the function into partial fractions. Leaving out of account any part which represents a rational integral function, we are then concerned with the sum of a finite number of fractions of the form A oA 1 \¢ @—a)? (a) (i=) a each of which we may, by 1, expand separately in a power series of the form - 2c, (2— 2)", provided z,# a. — This method enables us to see, moreover, that the radius of the resulting expansion will be equal to the distance of z, from the nearest point at which the denominator of the given function vanishes. 236. 412 Chapter XII. Series ot complex terms, Definition. For all real or complex exponents, the meaning to be attributed to the power e® is defined, without ambiguity, by the relation 2 ~2 PC esl bh hint: Tayi. And if p is any positive number, p? shall denote the value determined, without ambiguity, by the formula Em lo PpEi=e* gr, where log p is the (real) natural logarithm of p as defined in 3627 (For a non-positive base b, the power b% can no longer be uniquely defined; cf, however, p. 423.) As there was no meaning attached per se to the idea of powers with complex exponents, we may interpret them in any manner we please. Reasons of suitability and convenience can alone determine the choice of a particular interpretation. That the definition just given is a thor- oughly suitable one, results from formula 91, example 3 2% (leaving out of account the obvious requirement that the new definition must coincide with the old one for real values of the exponent); this formula was proved by means of a multiplication of series, the validity of which - holds equally for real and complex variables, and the formula must 237. 238. accordingly also hold for any complex exponent; it is e*fr.e?2 = est whence also pa p= ii pith, This important fundamental law for the algebra of powers therefore certainly remains true. At the same time it provides us with the key to the further study of the function eZ. 1. Calculation of ¢?. For real y's, we have = ln y2k41 TESTA ol Fht Se 1" eit Ft = Cos y-|isiny. 27 Jt may be noted how far removed this definition is from the elementary definition “z* is the product of % factors all equal to x”. — At first sight, there is no knowing what value belongs e. g. to 2%: yet this value is in any case uni- quely determined by the above definition. 28 By 234, 2, there can exist no other function than the function &® i defined which is regular in the neighbourhood of the origin and coincides on the real axis z=ga with the function ¢¥ defined by 838. For this reason we may indeed say that every definition of &* differing from the above would necessarily be unsuitable. § 55. The elementary analytic functions. — II. The exponential function. 413 Hence it follows that, for z =x 4174, er =e? ti =¢%.ciV =e" (cosy +} ¢siny). By means of this formula?? the value of ¢? may easily be determined for all complex z's. This formula enables us, besides, to obtain in a convenient and complete manner an idea of the values which the function e? assumes at the various points of the complex plane (in short, of : its stock of values). We note the following facts. 2. We have |e? =e%P =e? In fact Jom feos G8ing | VERT TET = 1, hence |e?|==|2%|- |e" | =e, because eZ > 0 and the second factor =1. Similarly, amel=J({ =v, also from the formula 38,1 just used. 3. ¢? has the periods 2 kn, that is to say, for all values of z, e? = eg?t2ni — prt2kai (k > 0, integral). For if we increase z by 2 m4 its imaginary part y increases by 2x, while its real part remains unaltered, and by 1. and § 24, 2, this leaves the value of the function unchanged. Every value which ¢? is able to assume accordingly occurs in the strip—a < (2) =v < 7, or in any A strip which may be obtained from it ; x by a parallel translation. Every such EL strip is called a period-strip; Fig. 11 represents the firstnamed of these strips. 4. e? has no other period, — in- 0 deed, more precisely: if between two special numbers z, and z, we have the relation ea =e, this necessarily implies that zg=2+2kmni. For we first infer that ¢%2-2z — 1; then we note that if e? = ei = g% (cos y | isiny) =1, 29 Duley: Intr. in Analysin inf. Vol. I, § 138, 1748. y 414 Chapter XII. Series of complex terms. we must by 2. have ¢?=1, hence z = 0. Further, we also have cosy 4-i¢siny=1 cosy==1, siny=0, hence y=2%n. Thus, as asserted, z2=2,—2,=2Fal. 5. e? assumes every value w == 0 once and only once in the period- strip; or: the equation e?=w,, for given w, 4= 0, has one and only one solution in that strip. If w, = R, (cos ®, + isin @,) with R, > 0, the number z,=1ogR iD, is certainly a solution of ¢?=w,, as ¢4 == gl8 Ra gi® = R (cos OD, | isin D) =w,. By 3., the numbers 2, +2kai (k=0,+1,+2,..) are also solutions of the same equation, and by 4. no other solutions 239. can exist. Now % may always be chosen, in one and only one way, so that —a 5 The latter lies between 1 and 2 for 2 every k==1,2,... (for it is = when 2 =1, and is less than this for every other 2, but > 1); therefore 2k — a¥p 1 J et @ “an Tw? and the radius of the cotseries = xz, by 94. Similarly that of the tan-series is found to be = 2. cotz and tanz have the period =. For cosz and sinz both change in sign alone when z is increased by nz. Here again we may show, more precisely, that cotz, =cotz, and tanz, = tang, 7 involve : Z,=u lta {.==0,%1,..). In fact, it follows from COS 2, COS Z, sin (z, — 2 cotz, —cotz,=—2— Ef == an =9) 2 sin z, sin 2, sin z, - sin z, that in the case of the first equation sin (z, — 2,) = 0, i.e. z, — 2, = ka. Similarly in the case of the second. 8. In the “period-strip”, i.e. in the strip — 2 <%= + 7s cotz and tanz assume every complex value w == +1 just once; the values w = + © are never assumed. To see this, write ¢?i2={_. The equation cotz = w then becomes Ll br 20 - OF. (=> 5 For each w <4 +4, { is a definite complex number == 0 and (by II, 5) there accordingly exists a 2 such that — za < J (¥) <=, for which ¢? ={, For f= — 42 we then have —-Z < 1 the inverse function of the exponential function e¢¥ — 1; i. e. substituting for y in YY yt 9] = 3] fess 33 Tt was precisely for this purpose that at the time we framed some of our inequalities in a form somewhat different from that required for the real domain (e.g. those mentioned on pp. 206—207, footnote 26). 242. 243. 420 Chapter XII. Series of complex terms. the above series and rearranging (as is certainly allowed) in powers of x, we reduce the new series simply to x. This fact — because it is purely formal in character — necessarily remains when complex quantities are considered. Hence, for every |z] = 1, e¥ —1=2 06 %=1-117z, if w denotes the sum of the series (L) md LD = =n We now adopt for the complex domain the Definition. A number a is said to be a natural logarithm of c, in symbols, : a=logc, if et =uc. In accordance with II, 5, we may then assert that every complex number ¢ &= 0 possesses one, and only one, logarithm whose imagin- ary part lies between — x exclusive and | xz inclusive (to the num ber 0, however, by II, 6, no logarithm can be assigned at all). This uniquely defined value will be more especially referred to as the prin- cipal value of the natural logarithm of c. Besides this value, there is an infinity of other logarithms of ¢, since with ¢% =¢ we have also eat2kni — ¢; thus if g is the principal value of the logarithm of ¢, the numbers a-t+2kni (k = 0, integer) must also be called logarithms of ¢. These values of the logarithm, apart from the principal value, are more specifically entitled subsidiary values.?* By II, 4 there are indeed no other logarithms of c. With these definitions, we may assert in any case that the above series (L) provides a logarithm of (14-2). But we may at once prove. more, namely the [3 Lad n—1 Theorem. The logarithmic series w = >) = n=1 point of the unit circle (including its rim, with the exception of the point — 1), the principal value of log (1-4 z). ~—— 2" gives, at each Proof. Writing z= r(cos ¢ + ising), we have SCD a : Siang» ny) el os qr x. 5 3(2 2 Je n= = ) » It is easy to see that this imaginary part varies monotonely from 0 Sade 1 We t- when 7 to the sum of the series sing — 34 If ¢ is real and positive, the principal value of logc¢ coincides with the (real) natural logarithm as formerly defined (36, Def.). § 55. The elementary analytic functions. —— VI. The inverse sine series. 42] increases from 0 to 1 and ¢ is kept constant®. In fact, denoting it for the moment by f(r), we have , = ; sin £0 = al Ht Spey nt em — +3(15) = Ia the denominator is constantly positive, so that f’(r) has the sign of sing, i.e. for every fized @ == 2m, it has a definite fixed sign for every 7 such that 0 <7 <1. This proves our last statement. The imaginary part, since it is 0 for » = 0, accordingly attains its ex treme values (positive or negative) for » = 1; its modulus is therefore never greater than the absolute value of Shor B22 Y Sem By 210, it lies, therefore, between — Z and += (both extremes ex- cluded) and the logarithmic series thus has for its sum, as asserted, the principal value of log (1 + z)*°. VI. The inverse sine series. We saw in II, 7 that the equation sinw = z, for a given complex z= 11, has exactly fwo solutions, — for z= +1 exactly one, — in the strip — 7 < R{(w) £ + nn. The two solutions (by III, 6) are sym- metrical, either with respect to +5 or ts accordingly, we may assert more precisely that the equation sin w = z, for an arbitrary given z (inclusive of +1), has one and only one solution in the strip J JT if the lower portions of its rim, from the real axis downwards, are omitted (cf. Fig. 12, where the parts of the rim not counted with the strip are drawn in dotted lines, and the others are marked by a continuous black line). This value of the solution of the equation sinw = z, which is thus uniquely defined for every complex z, is called the principal value of the function w=sin—1as. All the remaining values are contained, by III, 6, in the two formulac sin"1z + 2k, n—sinTlz242kan ‘and may be called subsidiary values of the function. 3 We may suppose @ + ka, since the content of the theorem is already established for the values of z excluded thereby (cf.the preceding footnote). 3 With the necessary care, this may also be inferred directly from the logarithmic series itself, by 98, 2. 422 Chapter XII. Series of complex terms. For real values of such that |2| <1, the series x8 | 1.3 y=o+g I represents the inverse series of the sine power series yy y=magy =the Exactly the same considerations as in V. for the case of the log- arithmic series now show that, for complex values of z such that |z| <1, the series 1-327 w=itg te 8 is the inverse series of the sine power series w— fee It therefore gives at any rate ome of the values of sin~1z. That this actually is the principal value, may be seen from the fact that, for 21 1, 2431, : ee 2 | (sin~22)| < [sin~2z| <2] +4150 + 22 ED Sie ay — a condition which the principal value alone fulfils. VII. The inverse tangent series. The equation tanw =z, as we know from IV, 3, has for every given z== +1 one and only one solution in the strip — x <%(w (exclusive) and -- = (inclusive). This remains true for every z= + on |z|=1, as well as for |z]| <1, and is proved as follows: | The sum w of the series (A), as may be seen by substituting the log-series, is 1 3 1 : w= grlog(1-}i2) — mrlog(l —i2) § 55. The elementary analytic functions. — VIII. The binomial series. 423 for every |z| <1, z 4% ¢, where principal values are taken for both logarithms. Accordingly, R (1) — + Slog (1 +12) — = Flog (1 — 72); by 243, both terms of the difference lie between — and + 5 hence RR (w) lies between —r and +=, the two extreme values being excluded in either case. Thus the series (A) certainly represents the principal value of tan™!z, provided |z| <1 and 24414, gq. e. d. VIII. The binomial series. To complete our present treatment of the special power series in- vestigated in the real domain, we have only to consider the bino- mial series a fC n 0+) = >t )a n=0\" in the case where the quantities occurring there — i. e. the exponent as well as the variable x — assume complex values. We start with the Definition. The name of principal value of the power b®, where 244. a and b denote any complex numbers, with b==0 as the only condi- tion, 1s given to the number uniquely defined by the formula 3% g2108b : when log b is given its principal value. — By choosing other values of log b, we obtain further values of the power, which may be called its subsidiary values. All these values are contained in the formula BY ploguthag each value being represented exactly once, if logd is given its prin- cipal value and k takes all integral values 20, Remarks and Examples. 1. A power b? accordingly has an infinite number of values in general, but possesses one and only one principal value. 2. The symbol i%, for instance, denotes the infinity of numbers (all real numbers, moreover) i(3E+2hai) —Z op eilogi+2kai) =e 2 vo (=0, £1,402, ..) TT of which e¢ 2 is the principal value of the power if, 3. The only case in which a power b% will not have an infinite number of values is that in which gird *=0,+1,12... 424 Chapter XII. Series of complex terms. gives only a finite number of values; this will occur if, and only if, %.a assumes, for 2 =0, +1, + 2,..., only a finite number of essentially different values. Here two numbers are described (just for the moment) as essentially different if, and only if, they do not differ merely by a (real) integer. Now this is the case if, and only if, a is a real rational number, as may be seen at once; and the number of “essentially different” values which may in this case be assumed by k-a is given by the smallest positive denominator with which @ may be written in fractional form. 3 4. It follows that b™ =V3, where m is a positive integer, has exactly m different values, one of which is quite definitely distinguished as the prin- cipal value. 5. The number of different values of 5% will reduce to one, by 3. and 4., if, and only if, a is a real rational number of denominator 1, i. e. a real integer. For all real integral exponents (but for these alone), the power thus remains now as before a single-valued symbol. 6. If b is positive and a real, the value formerly defined (v. 833) as the power b% is now the principal value of this power. 7. Similarly, the values defined in 286 for e? and p? (p > 0), are now, more precisely, the principal values of these powers. In themselves, these sym- bols would represent, for complex values of z, an infinity of values, in ac- cordance with our last definition. Nevertheless, we shall keep in future to the convention that €% and generally p® for any positive p, shall represent the value defined by 286, 1. e. the principal value only. 8. The following theorems will show that it is consistent to define 5% also for b=0 when $R (¢) > 0. The value attributed to the power in that case is 0 (uniquely). After making these preliminary preparations, we proceed to prove the following far-reaching 245. Theorem. For any complex exponent « and any complex z in |z| <1, the binomial series bd o a o - o 9 o n (5) ma (1)e (Soot (2) converges and has for sum the principal value of the power {UL a)se Proof. The convergence follows word for word as in the case of real z’s and o's (v. pp. 209—210), so that we have only to prove the statement as to the sum of the series. Now for real 2's such that || <1, and real «'s, we may substitute pr-1 tS a" = alog (1+ 2) 37 Abel: J. f. d. reine u. angew. Math., V. 1, p. 311. 1826. § 55. The elementary analytic functions. — VIII. The binomial series. 425 x 2 for y in the exponential series ¢¥ =1- y a —+--. and so obtain, after rearranging in powers of x (allowed by 104), the power series for ¢#logd+d = (1 -} 2)? i e. the binomial series 2%) Let us proceed in this manner, purely formally in the first instance, assuming « complex and writing z for x; i. e. we substitute w=g. 5 Em in ev = 32 n=1 n=1 and rearrange in powers of z. We necessarily obtain — without refer- ence as yet to any question of convergence — the series = aN 26) whose sum would therefore be proved to be g*108(1+2) — (1 4 2)* (where the principal value is taken for the logarithm and hence for the power also), if we could show that the rearrangement carried out was per- missible. Now by 104, this is certainly so; in fact the exponential . . —1)2—2 : series converges everywhere and the series o >) kel z" remains convergent for | 2] < 1 when « and all the terms of the series are replaced by their absolute values. This proves the theorem in its full extent. If we split up (14 2)“ into its real and imaginary parts, we obtain a formula due to Abel, which is complicated in appearance, but which for that very reason shows how farreaching a result is contained in the preceding theorem, and from which we also obtain a means for evaluating the power (1-}z)% Writing z= 17 (cos ¢ + 4 sing) and e=p-+1iy, 0 0, to a finite sum, and has then the (ipso facto unique) value (1+ 2)%; in particular for «= 0 it has the value 1 (also when z= — 1). If a does not have one of these values, the series con- verges absolutely for |z| <1 and diverges for |z| > 1, while it exhibits the following behaviour om the circumference |z|=1: a) if R(«)>0, it converges absolutely at all points on the circum- ference; : b) if R(«) £ — 1, it diverges at all these points; ¢) if —1< R(x) L0, it diverges at z—=—1 and converges con- ditionally at every other point of the circumference. The sum of the series -when it converges is invariably the principal value of (1+ 2)“; in particular, its value is O in the case z= —1. Proof,” Writing (— ore) =g .;5 we have n Sngr (%) he Le] alae], ne rz ig > = 1) hence theorem 229 may be applied, and the validity of a), b) and c) follows immediately. Only the case of the point z = —1, i.e. the con- vergence of the series SE n requires special investigation. Now =e} r1 rer tii=tnal so 1- (+5) - (= -0fi- = pei § 55. The elementary analytic functions. — IV. The binomial series. 427 and in general, as may at once be verified by induction: p(B (2 tC) Sv Ten sd) the partial sums of our series are equal to the partial products, with the same index #, of the product 17 (1 — 2)- The behaviour of this n=1 product is immediately evident. In fact: 1 ¥ %la)=p4>0, choose B' such that O0< m, hence (1- 2)(1- 2) (=| <= =a (1 =) By 126, 2, it follows at once that the partial products, and hence the partial sums of our series, tend to 0. The series therefore converges to the sum 0 3°. 2. If, however, R(a)= — f <0, we have o B 1-2 eehs, whence it again follows by multiplication that 0-9-5) (=3|> (+9 a+3)-0+3. and hence that the left hand side tends to oo. The series therefore diverges in this case. 3. If, fmally, RNe)=0, a=1iy, say, with yX0," the sth partial sum of our series is W+in(t+)-- (14+). The fact that this value tends to no limit as # — -- co may be proved most speedily in the present connection as follows: On account of the ab- Ly \ 2 solute convergence of the series 32) , we have, by § 29, theorem 10: (+7) (+) (1+) Letting # — -}- co, the right hand side evidently tends to no limit; on the contrary, the points which it represents for successive values of » circulate incessantly round the circumference of the unit circle in a constant sense, the interval between successive points becoming smaller ; 1 i ty [++ +) o 3 The mere convergence of 3 (— 1)» a follows already from 228 and we see that the convergence is absolute when Ri («) > 0. It is the fact of the sum being 0 which requires the artifice employed above for its detection. 428 Chapter XII. Series of complex terms. and smaller at each turn. In view of the asymptotic relationship, the same is therefore true of the left hand side. Hence our series Z00(7) also diverges when R(«)=0. Thus theorem 247 is established in all its parts, the behaviour of the binomial series is de- termined for every value of z and of ¢, and its sum for all points of convergence is given by means of a “closed expression”. § 56. Series of variable terms. Uniform convergence. Weierstrass’ theorem on double series. The fundamental remarks on series of variable terms = 21.2) n=0 are substantially the same for the complex as for the real domain (v. § 46); but instead of the common interval of definition we must now assume a common region of definition, which for simplicity — this is also quite sufficient for most purposes — we shall suppose to be a circle (cf. p. 403, footnote 17). We accordingly assume that 1. A circle |z2 — z,| 0, it is possible to choose a single number N > 0 (independent, therefore, of z) such that [fain DF fra =r, 0] <'e for every nm > N and every z in the circle considered. Remarks. 1. Uniformity of convergence is here considered relative to all the points of an open or closed circle4?. Of course other types of region or indeed arcs of curves or any other set 9t of points, not merely finite in number, may be taken as a basis for the definition. The definition remains the same in sub- stance. — In applications, we shall usually be concerned with the case in which the terms f, (2) are defined, and the series Xf, (2) converges, at every point interior to a circle |z—2z,| < 7 (or a domain (), but the convergence is uniform only in a smaller circle |z— z,| 0 be chosen, to be kept fixed throughout. By hypothesis, we can determine a k, such that, throughout |z — z,| < 0, jsp — st < d= orem, for every k such that k’ > k > k,, if we write 5=5,8=E@ +" +E. Now the function sp (2) — s,(2) is a definite power series, whose mth coefficient is (k+1) (k+2) visi (%') Alert Lain he Lgl, By Cauchy's inequality 238, we therefore have 14 i 4 e / | a+ q+) LL |g) k, on the same circumference | 5, (2)] = | Stor1(2)| + | 8: (2) — st+1(®)| EM + =M. Again, using Cauchy's inequality we obtain, for every n=0,1, 2,..:, \ M jo = 7.2 2 ot ol 7.9) =u whatever the value of k. Hence M | 4. = o" > and 34 (2 — z,)" therefore converges for | z — z,| < ¢. Since the only n=0 restricion on @ was that it should be <7, the series must even converge for |z—z,| <7. (In fact, if z is any determinate point satisfying the inequality |z — z,| < 7, it is always possible to assume o to be chosen so that |z—z,| <9 0. We can determine k, so that, in ns J) |Z 0; ; sr — sp] < & =o BE for every k such that > k > k,. By Cauchy's inequality, it follows as before that, for 2 > k > k, and for every n > 0, fo Ad eg | al+D | gled L.... + a #1] = a Making &' — + co, we infer that, for every k > k, and every n> 0, j4,— (2.2 + 2, + TR + a,?)| 0 and @’ << 9 < » have been chosen arbitrarily, we can determine % so that £ |B — Sli k, and every | z — z,| < ¢’. This implies, however, that for these values of z » : Pilg= 27.12}; Lie see). The numbers po’ and p were subjected to no restriction other than 0 k,, Pe Sw sei refs. ] = RE Be AR = 0 0? (e—¢)? Hence for those values of z — indeed, by the same reasoning as before, for every |z —z)| 207) (=1,2,..., fixed) for every |z — z,| <7; i e. the series Xf, (z) obtained by differen- tiating term by term, » times in succession, converges in the whole circle |z — z,| < » and gives the »th derived function of F(z) there. =&€. Remarks. 1. A few examples of particular importance will be discussed in detail in the next section but one. 2. The fact of assuming the convergence uniform in a circular domain is immaterial for the most essential part of the theorem: If the terms of the series 2 fj (2) are analytic in a domain of arbitrary shape?® and if every point z, of the domain is the centre of a circle |z—z,| < p (for some g) which belongs entirely to the domain and is a circle of uniform convergence of the given series, then this series also represents a function F(z) analytic in the domain in question, whose derived functions may be obtained by differentiation term by term. — Examples of this will also be given in § 58. 46 Cf. p. 403, footnote 17. 4834 Chapter XII. Series of complex terms. § 57. Products with complex terms. The developments of Chapter VII were conducted in such a way that all definitions and theorems relating to products with “arbitrary” ‘terms hold without alteration when we admit complex values for the factors. In particular the definition of convergence 128 and the theo- rems 1, 2 and 5 connected with it, as well as the proofs of the latter, remain entirely unchanged. There is also nothing to modify in 127, the definition of absolute convergence, and the related theorems 6 and 7. On the other hand, some doubt might arise as to the literal trans- ference of theorem 8 to the complex domain. Here again, however, everything may be interpreted as “complex”, provided we agree to take log (1+ a,) to mean the principal value of the logarithm, for every sufficiently large #. The reasoning requires care, and we shall therefore carry out the proof in full: 250. Theorem. The product II(1 + a,) converges if, and only if, the series, starting with a suitable index m, 3 log(1 +a), n=m+1 ’ whose terms are the principal values of log(1+ a,), converges. If L is the sum of this series, we have, moreover, 1 +a)=Q0+a)1+a)...(1+a,) en » Proof. a) The conditions are sufficient. For if the series 3 log (1+ a,), with the principal values of the logarithms, is con- n=m+1 vergent, its partial sums s _, (n > m), tend to a definite limit L, and consequently, since the exponential function is continous at every point, ; £5 m= (1 + 200s) (1 ft Ap +2) A (1 =} a2,)—> ok i. e. it certainly tends to a value #0. Hence the product is con- vergent in accordance with the definition 125 and has the value stated. b) The conditions are necessary. For, if the product converges, given a positive ¢, which we may assume < 1, we can determine 7, so that £& |(1 + ans) + Opie) (A+ a) — 1 < °T for every n>mn, and every k =1. We then have, in pargeniag ja, |< o =< 4 for every n > m,, and the inequality | a, | < 5 is thus certainly fulfilled for every n greater than a certain index m. We may. now show further that for the same values of # and k (using the § 57. Products with complex terms. 485 principal values of the logarithms) *? n+k (a) > log(1+a)l n,, we also have, for these values of », (b) [log(1+a,)| n, and every k>1. Accordingly, for some suitable integer ¢*% we certainly have (©) |log 1 +a, ,,)+log (1a, Yt log{l ta, .) 42 EA <é¢ and it only remains to show that ¢ may in every case be taken = 0 Now if we take any particular # > n,, this is certainly true for k =1, by (b). It follows that it is true for 2 = 2. For in the expression log1+4a,.,)+log(l+a,.,)+2qai the modulus of either of the two first terms < ¢, by (b), and by (c) the modulus of the whole expression has to be <¢g; as <1, g cannot, therefore, be an integer different from 0. For corresponding reasons, it also follows that for k = 3 the integer ¢ must be O, and this is then easily seen by induction to be true for every %k. This establishes the theorem. The part of theorem 127, 8 relating to absolute convergence may also be immediately transferred to the complex domain, — viz. the series 5 log (1+ a,) and the product Ir (14a) n=m+1 n=m+1 are simultaneously absolutely or non-absolutely convergent, in every case. Similarly the theorems 9—11 of §§ 29 and 30 remain valid. In fact, it remains true for complex a's of modulus < that in log (1 + a,) = a, + 2, a,’ 47 The logarithms are always taken to have their principal values in what follows. 4%: In fact, for zl << Eds | 2] log (+9 lz] +54 Sal + 24 = oS 200). 49 For the principal value of the logarithm of a product is not necessarily the sum of the principal values of the logarithms of the factors, but may differ al 2 from this sum by a multiple of 2x4. Thus e. g. log: = , but log (¢-i-1+1) =log1=0, if we take principal values throughout. 251. 436 Chapter XII. Series ot complex terms. : the quantities ©, are bounded, — since when |z| < > og(+2=2+[—g+t—T+—]2 while the expression in square brackets clearly has its modulus < 1 for those z's. Finally, the remarks on the general connection between series and products also hold without alteration, since they were purely formal in character. Examples. SEN ; Tet LJ] (1 Sr >) is divergent. For 2 |a,|2 = SIE is convergent, so that by § 29, theorem 10, the partial products : 5 > ; 1 T 7 7 7 til t—=a... + = mm meet alin ie 1 — ( 2 3 > fasliseplft anon rid ne i the right hand expression represents, for successive values of #, points on the circumference of the unit circle, which circulate incessantly round this circum- ference at shorter and shorter intervals. ¢, therefore tends to no limiting value. (Cf. pp. 427—38.) © pm-+1)+ (1+1) i 2 I ernra— "= i . In fact, the =n! partial product is at once 1+ (n+1)¢ : seen to ere TT , which —-—1. od 8: For {2lf,(2),... are all ana- 52. lytic at least in the (fixed) circle |z — zy| < 7; if, further, the series 2G converges uniformly in the smaller circle |z— z,| < 0, for every positive 0 < 7; then the product II(1 +f, (2)) converges everywhere in |z — zo] <7 and represents a function F(z) which is itself analytic in that circle. The proof follows the same line of argument as that of the continuity theorem 218,1 almost word for word. To establish the con- vergence and analytic character of the product at a particular point z, in the circle |2—2,| <7, we chooseag for which |7, — 2, ( m and every |z— z,| <0; then for all these u's and z's, [Pu @ =| AF Fos @)- (1H Fo @) | SV ma @ 1 +k 1000] < 3, It follows precisely as on p. 382 that the series SE + Duets re Dts) tr Toe + (Pn I Dad) + ne converges uniformly in |z — z,| < 0. As all the terms of this series are analytic in |z — z,| <7, the series itself, by 249, therefore re- presents a function F, (z) analytic in |z — z,| << 0. Hence F@=H0+E)=0+AE) + ful) Fuld is also an analytic function, regular in that circle and at the point z, in particular. Now z; was chosen arbitrarily in | 2 ee Zoi <7; thus F (2) is analytic in the whole circle |z — z,| <7, q. e. d. From the above considerations, we may deduce two further theor- ems, which provide an analogue to Weierstrass’ theorem on double series: Theorem 1. With the assumptions of the preceding theorem, the 253. expansion in power series of F(z) may be obtained by expanding the product term by term. More precisely, we know that the (finite) product k P@) = (1+1,6) may be expanded in a power sevies of centre z, which converges for |z2 — z,| < 7, since this is the case with each of the functions firifar sve . 15% 438 Chapter XII. Series of complex terms. Let the expansion be P(x) = AP + A(z — 2) + ie — 2) APE — 2) Then for each ied)» n=0,1,2,..., the Limit lim AP = =A, E>+» exists, and Bly= ie hif)=J 34 = 2) Prool., By $ 16, theorem 2, the uniform convergence, in |2— 2,| Zo, of the series Pmt1 ~~ £m — Divs) — ery used in the preceding proof, implies the uniform convergence in the same circle of the series 2042.02 + 450 —-P a2 Applying Weierstrass’ theorem on double series to this series, we obtain precisely the theorem stated. Finally we prove a theorem about the derived function of F(z), quite similar to 218, 2: Theorem 2. For every z tn |z— z,| <7 for which F(z) <0, we have FF’ (2) eS fn r@ FE = +E i. e. the series on the right hand side converges for all these values of z and gives the ratio on the left hand side, the logarithmic dif- ferential coefficient of F(z). Proof. We saw that the expansion F(2)= Py (2) + (Py (2) — Py (2) +++ was uniformly convergent in |z —z,|

Ay «Gry eens LiL m = 2p. Then - LY. fp) fit Yo (140-2) ut oms{i) (10 (ra) (0 2) fos =z(1+5) (142 thee {xd 1 AE wht) Seed : where log (1 +Z) = Zl As | Z| <3 (cf. p-435) we have |, |< [2/2 < 03, | pe and the last factor in the preceding expression therefore remains m. Similarly the last factor but one (see p.295), also ‘remains less than a fixed number 4,. As the remaining factor is also always less than a fixed number 4; for every |z| m. On the other hand, the first m functions |g, (2)|, 8?) |, ..., | gm(?) | also remain bounded for every |z| < po; the existence of the number 4 as asserted in the text is thus established. If z is restricted to lie in a circle &, in the interior and on the boundary of which z+0,—1,—2,... and |z| m » Nm+1 Nn 1 1 1 = : 1 : 2k —tonn) ; , m+12 ne 2. @) ( 2 ( 2) 2 2 Ee — 1+ 1 1+ = From this we infer in exactly the same way that a constant 4’ exists such : that 1 i 0 | <4 in @,forevery n=1,2,...4 Tn 1 J a a § 58. Special classes ot series of analytic functions. — A. Dirichlet’s series. 441 every |z| 4 and diverges when R (2) < 4. The number i may also be — co or - 00; in the former case the series converges everywhere, in the latter nowhere. Further, if Ad+o0o and X>1, the series is uniformly convergent in every circle of the half-plane R(z) => 2’ and accordingly the series, by Weier- strass’ theorem 249, represents a function analytic in the half-plane ee The proof follows a line of argument similar to that used in the case of power series (cf. 98). We first show that if the series con- verges at a point z,, it converges at every other point z for which R (2) > R(z,). As however 2 1 JET rE it suffices, by 184, 3a, to show that the series — Em ye -—] > ni 417% rey ip HR 2 | +2 | 5 More generally, a series is called a Dirichlet series when it is of the form = %n_ or of the form ST e~’7% where the PA's are positive numbers and the i s amy real numbers increasing monotonely to + 00. % The existence of the half-plane of convergence was proved by LL.W.Y, Jensen (Tidskrift for Mathematik (5), Vol. 2, p. 63. 1884); the uniformity of the convergence and thereby the analytic character of the function represented were pointed out by E. Caken (Annales Ec. Norm. sup. (8), Vol. 11, p. 75. 1894). 442 Chapter XII. Series of complex terms. is convergent. Writing (for a fixed exponent (z — z,)) 1 2—2 Pa (x 3% =) =14 2, the numbers & — (z — z,)®® and are therefore certainly bounded; |#,| <4, say. The nth term of the above series is therefore A = TF We—z * and the series is accordingly convergent when %(z —z,) > 0. As a corollary, we have the statement: If a Dirichlet series is divergent at a point z=1z,, it is divergent at every other point whose real part is less than that of z,. Supposing that a given Dirichlet series does not converge everywhere or nowhere, the existence of the limiting abscissa 4 is inferred (as in 93) as follows: Let 2 be a point of divergence and 2” a point of convergence of the series, and choose z,<< R (2) and y,>R(?"), — both real. For z=12x, the series will diverge, for z=1y, it will converge. Now apply the method of successive bisection, word for word as in 93, to the interval Jo==%,.-. 9, on the real axis. The value 1 so obtained will be the required abscissa. Now suppose A’ > 1 (for = — oo, A’ may therefore be any real number); if z is restricted to lie in a domain G in which % (2) > 1’ and |z| < R, — so that in general G will take the shape of a seg- ment of a circle, — our series is uniformly convergent in that domain. To show this, let us choose a point z, for which 1 < R(z,)) <4’; as before, we write ay , 1 n? = no pi sags ul 5% More generally, we may at once observe that if |z| < ° and |w |< R, and if we write, taking the principal value, A+2)¥=1+z2w+ 3-22 the factor ©, which depends on z and w, remains less than a fixed constant for all the values allowed for z and w. — Proof: a +2)¥ = ewlog(1+2) i WE +12?) with = eb z a 7% = = ’ y= + vee 2 3 4 For every jefe we therefore have | |< 1; hence in MEE oy pug ( 499+ TE THIN 2(1-Lyz)2 3(1Lyz)3 Pin Lig yas 22 ay : oO. =14+wz+ E 7 + the expression in square brackets, which was denoted by ¢}, satisfies the in- equality [8] < £25, This is at once obvious if we replace all the quantities in the brackets by their absolute values and then replace || and |z| by 1 and finally [w| by R. Ra § 58. Special classes of series of analytic functions. — A. Dirichlet’s series. 443 > is a convergent series of constant terms; by 198, 3a it there- n 0 fore suffices to show that el bo 1 n?"% (n+ BT converges uniformly in the domain in question and that the factors n=1 ——— are uniformly bounded in G. Now, writing /'— % (z,)=0 (> 0), nn? %o 1\%—% . i — 10. (1+) Using the evaluation given in the preceding footnote (or else directly, by 1 expanding (1 += a (e+) see that a constant A certainly exists such that the difference within the modulus signs on the right hand side of the above fnequality is in absolute value 1 1 n?%o = (mn 1)2"% 1 = in powers of (z—z,) we now A oS for every z in our domain and every n=1,2, 3, ...." The ‘whole expression on the right is thus 4 < plto On the other hand, since lL. the factors bi are — pn? 720 uniformly bounded in G. By 198, 3a, this proves that the Dirichlet series is uniformly convergent in the domain stated, and hence, in particular, that every Dirichlet series represents a function which is analytic in the interior of the region of convergence of the series (the half-plane % (2) > 1). From 1 né"%o ap an 2 = n? n?o it follows at once that if a Dirichlet series converges absolutely at a point z,, it does so at any point z for which (2) > R(z,), and if it does not converge absolutely at z,, then it cannot do so at any point z for which (2) < R(z,). Just as before we obtain Theorem 2. There exists a definite real number 1 (which may also be +00 or — oo) such that the Dirichlet series converges ab- solutely for R(2) >1, but not for R(z)<< 1. Of course we have 4 R(z,) +1, then neo > In is absolutely convergent, for |%&|—|fn|__1 _ im n’ y g 2 n? 720 2 @—20) R (2 — 2, >1. This proves the statement at once. Remarks and Examples. 1. If a Dirichlet series is not merely everywhere or nowhere convergent the situation will in general be as follows: the half-plane % (z) << 2 of diver- gence of the series is followed by a strip A < /, the series converges absolutely. 2. It may be shown by easy examples that the difference / — 1 may assume any value between 0 and 1 (both inclusive), and that the behaviour on the bounding lines $f (2) =4 and R (2) =! may vary in different cases. 3.. The two series > 2 ; : ar and oF provide simple examples of Dirichlet series which converge everywhere and nowhere. 1 : : 4. Nr has the abscissa of convergence A=1; thus it represents an n analytic function, regular in the half-plane ® (2) > 1. It is known as Riemann’s {-function (v. 197, 2, 8) and is used in the analytical theory of numbers, on account of its connection with the distribution of prime numbers (see below, Rem. 9)5% 5. Just as the radius of a power series can be deduced directly from its coefficients (theorem 94), so we may infer from the coefficients of a given Divichlet series what positions the two limiting straight lines occupy. We have the following Theorem. The abscissa of convergence J of the Dirichlet series Fin is n invariably given by the formula = 1 A= lim LEN ee z>+w T where x increases continuously and u=el?, v=[e7]. Substituting | a, | for a, in this formula, we obtain I, the limiting abscissa of ab- solute convergence bS. 6. A concise account of the most important results in the theory of Dirichlet's series may be found in G. H. Hardy and M. Riesz, Theory of Divichlet's series, Cambridge 1915. 57 A detailed investigation of this remarkable function (as well as of arbitrary Dirichlet series) is given by E. Landau, Handbuch der Lehve von der. Verteilung der Primzahlen, Leipzig 1909, 2 Vols. 88 As regards the proof, we must refer to a note by the author “Uber die Abszisse der Grenzgevaden einer Divichletschen Reihe” in the Sitzungsberichte der Berliner Mathematischen Gesellschaft (Vol. X, p. 2, 1910). § 58. Special classes of series of analytic functions. — A. Dirichlet’s series. 445 7. By repeated term-by-term differentiation of a Dirichlet series F (2) =F, we obtain the Dirichlet series 0 1 vy yr S adorn (fixed 7). n=1 n? As an immediate consequence of Weierstrass’ theorem on double series, these necessarily all have the same abscissa of convergence as the original series, and represent, in the interior of the half-plane of convergence, the derived functions F® (3). 8. By 285, the function represented by a Dirichlet series can be expanded in a power series about any point interior to the half-plane of convergence as centre. The expansion itself is provided by Weierstrass’ theorem on double @0 series. If, for instance, it is required to expand the function { (2) = 3 — about z, = 2 as centre, we have for 2 =2, 3, 1 1 1 1 Sa 1x = log k)" 2 SE =r (z pogo 3 (1m CE gps (k fixed), and this continues to hold for k =1 provided we interpret (log 1)? as having the value 1. Hence for » =>0 LD a geghe ge - kh? (n fixed), Ms k=1 which gives the desired expansion LE) = ni Ely [ 50 (log 2)" e-2r] = 5 (2 Eh) etme 9. For R&D >1, the series 3 x and the product [| n=1 n? ; 1 (where p takes for its values all the prime numbers 2,38,5,7,... in succession) have everywheve the same value, and accordingly both represent the Riemann €-func- tion { (2). (Euler, 1737; v. Introd. in analysin, p. 225.) Proof. Let z be a definite point such that R(z2)=1+4+06>1. By our remark 4 and 127, 7, the series and product certainly converge absolutely at this point. We have only to prove that they have the same value. Now 1 1 1 1 HR EY multiplying these expansions together, for all prime numbers p < N, — where N denotes an integer kept fixed for the moment, — the (finite) product so obtained is « FuLoy frid PEN — Pei my We where the accent on the X rile that only some, and not all, of the terms of the series written down are taken. Here we have made use of the elemen- tary proposition that every natural number Z>2 can be expressed in one and only one way as a product of powers of distinct primes (provided only positive 258. 446 Chapter XII. Series of complex terms. ‘integral exponents are allowed and the order of succession of the factors is left out of account). Accordingly 1 $1 1 PEN 1-—p7 n=1"7 gp A, nlts On the right hand side we have the remainder of a convergent series, which tends to 0 when N—-4 00. This proves the equality of the values of the infinite product and of the infinite series, as was required. 10. By 287, we have for RR (2) >1 1 1 i o © (1) pg HO- += g-2)- 2 £2) II pa I Pp? 2 n? where nM=1, p@=—1, p@=-1, a@=0, pG)=—1, p@®)=+1,... and generally u (n)=0, +1 or —1 according as = is divisible by the square of a prime number, or is a product of an even number of primes, all different, or of an odd number of primes, all different. This representation also shows that for ® (2) > 1, we have always { (2) &= 0. The curious coefficients u(r) are known as Mobius’ coefficients. There is no superficial regularity in the mode of succession of the values 0, +1, —1 among the numbers yu (n). 11. Since (= Sd converges absolutely for Rf (2) > 1, we may form n the square (£(2))® by multiplying the series by itself term by term and re- arranging in order of increasing denominators (as is allowed by 91). We thus obtain sos N22, ke AE where 7, denotes the number of divisors of n. — These examples may suffice to explain the importance of the (-function in problems in the theory of numbers. B. Faculty series. A faculty series (of the first kind) is a series of the form = nla, (F) = 2+) ...(z+n)’ which of course has a meaning only if 240, —1, — 2,.... The questions of convergence, elucidated in the first instance by Jensen, are completely solved by the following Theorem of Landauw®®. The faculty series (F) converges — with the exclusion of the points 0, — 1, — 2, ... — wherever the “asso- ciated” Dirichlet series 5 : a n=1"7 converges, and conversely the latter converges wherever the series (F) con- verges. The convergence is uniform in a circle for either series, when it is so for the other, provided the circle contains none of the points 0, —1, — 2, ... either in its interior or on its boundary. % Uber die Grundlagen der Theorie der Fakultitenreihen. Miinch. Ber, Vol. 36, pp. 151—218. 1906. § 58. Special classes of series of analytic functions. — B. Faculty series. 447 Proof. 1. We first show that the convergence of the Dirichlet series at any particular point £0, — 1, — 2, ... involves that of the faculty series at the same point. As n! a, ag 1 EHD GED) af B® if g (z) has the same significance as in 254, example 4, it is sufficient, by 184, 3a, to show that the series e lL... i G | 81 (2) — 8a (2) | net1|8® E410) nei | 8) utr] is convergent. Now : a tends to a finite limit as # increases, namely n to the value I'(z); hence, in particular, this factor remains bounded for all values of n (z being fixed). Hence it suffices to establish the con- vergence of the series PIFORT0 But this has been done already in 254, example 4. 2. The fact that the convergence of the faculty series at any point involves that of the Dirichlet series follows in precisely the same manner, as again, by 184, 3a, everything turns on the convergence of 2g.) —g,+.:.@] 3. Now let ® be a circle in which the Dirichlet series converges absolutely and which contains none of the points 0, —1, — 2,..., either as interior or boundary points. We have to show that the faculty series also converges uniformly in that circle. By 198, 3a, this again reduces to proving that A Ent1(2) — 8: (2) | n=1 En (2) 8n41(9) | is uniformly convergent in ® and that the functions 1/g,(z) remain uniformly bounded in &. The uniform convergence of Zen @—e.)] was already established in 254, 4. Also it was shown on p. 440, footnote 53, that there exists a constant A’ such that 1 ’ Eo <4 for every z in ® and every un. This is all that is required. (Cf. § 46, theorem 3.) 4. The converse, that the Dirichlet series converges uniformly in every circle in which the faculty series does so, follows at once by 198, 3a from the uniform convergence of the series 2'|g, ,, (2) — g,(2)] and the uniform boundedness of the functions g, (z) in the circle, both of which were established in 254, 4. 448 Chapter XII. Series ot complex terms. : Examples. 1. The faculty series 2 1 n! = 22 t1z(z41)... (24+ n) converges at every point of the plane £0, —1,.... For the Dirichlet series RN 1 i 2 gnt+l nz is evidently convergent everywhere. As 1-1 iva 1 #7 = wtl El)’ ! i i a i wl. oe Bl zx z@x4+1)(x+2) z+ z(etl)... @F) the given faculty series results simply, by Euler's transformation 144, from the series & 17" 1 1 1 + z+ 2 PE z-+1 —-L 2. It is also easily seen (cf. pp. 265—6) that for R (2) > 0 1 0 1! (n—1)! Eerie Dern Tere nT eu ee Thin 2@+1)- En)’ To show this, we have only to subtract the terms of the right hand side suc- cessively from the left hand side. After the »'® subtraction we have 1 22 n! 1 nln? 2+) E+2)...¢+n) zn? 2@+1)...@z+n)’ and this, by 254, example 4, tends to 0 when # —» oc, provided R (2) > 0. (Stirling: Methodus differentialis, London 1730, p. 6 seqq.) C. Lambert’s series. A Lambert series is a series of the form © zn 7 60 > GF lf we again inquire what is the precise region of convergence of the series, it must first be noted that for every z some positive integral power of which is 1, an infinite number of the terms of the series become meaningless. For this reason, the circumference of the unit circle will be entirely excluded from consideration ® while we discuss the 60 A more extensive treatment of this type of series is to be found in a paper by the author: Uber Lambertsche Reihen. Journ. f. d. reine u. angew. Mathem. Vol. 142, pp. 283—315. 1913. 81 This does not imply that this series may not converge at some points z, of this circumference, for which z”++1 for every » = 1. This may actually happen; but we will not consider the case here. - § 58. Special classes of series of analytic functions. — C.Lambert’s series. 449 question of convergence of these series, and the points inside and outside the circle will be examined separately. We have the following theorem, which completely solves the question of convergence in this respect: Theorem. If Xa, converges, the Lambert series converges for every z 359. whose modulus is £1. If Za, is not convergent, the Lambert series converges at precisely the same points as the “associated” power series Sa, 3" — provided |z| +1 as before. Further, the convergence is uniform in every circle ® which lies completely (circumference included) within one of the regions of convergence of the series and contains no point of modulus 1. Proof. 1. Suppose Za, divergent. The radius » of Xa, 2" is in that case necessarily <1 and we have to show first that the Lambert series and the associated power series converge and diverge together for every |z| <1, and that the Lambert series diverges for |z|> 1. Now a = Fat —-(1—2") a 2a ts TIE g= Na. 7 Lh Accordingly, it suffices, by 184, 3a, to establish the convergence of the two series Z| A—t)— (=) = Z| — or = [1-2] Z| 2 and 1 1 >To far] TF IZ =11—=|- Zama en for |z | <1. The first of these facts is obvious, however, while he second follows from the remark that for |z| <1, we have |1—2" | =4 — for all sufficiently large #’s. On the other hand, if the Lambert series converged at a point zg, where |z,| > 1, the power series an n Sits would converge for z =z,, and by 93, theorem 1, would have also to converge for z = + 1. Hence the series an Fra Ja, Zo" = Ja, would also have to converge, which is contrary to hypothesis. Finally, the fact that the Lambert series converges uniformly in |z] < 0 < » may at once be inferred from the corresponding fact in the case of the power series X|a, 2" |, by § 46,2, — in virtue of the inequality a Z ny] gn Ce 450 Chapter XII. Series of complex terms. The case where Xq diverges is thus completely dealt with. 2. Now suppose 2a, convergent, so that 3g 2" has a radius #=>1. The Lambert series is certainly convergent for every |z| <1 and indeed uniformly so for all values of z such that |z|

1, we have 3 = <5< 1, this reduces the later assertions to the pre- zn STE Sy 2a, 1-— and as ceding ones, and the theorem is therefore established in all its parts. By the above, a very simple connection exists, in the case where 2a, is convergent, between the sum of the series at a point z outside the unit circle and the same sum at the point 5 inside it Accordingly it will suffice if we consider only that region of convergence of the series which lies inside the unit circle. This is either the circle |z| < 7 or the unit circle |z| < 1 itself, according as the radius 7 of the series 2'¢ 2" is <1 or => 1. Let 7, denote the radius of this perfectly definite region of convergence, The terms of a Lambert series are analytic functions regular in |z| <7,, and for every positive gp < 7,, the series is uniformly con- vergent in | z| < p: hence we may apply Weierstrass’ theorem on double series to obtain the expansion in power series of the function re- presented by a Lambert series in |z| <7,. We have : Gi =—mzta,d tata tata ta + 2 2 2 4 6 DR ay 2 + a,z +a,z sl wis 78 r aS 3 6 Asie, Gig = a; Zz a4 ; ‘ “ra ms a zt + 41 _24 1 oi 0 el elie eee Te ee WL MC and we wy add all these series yoettiol orm by term. In the pin row, a given power 2" will occur if, and only if, » is a multiple of &, or k a divisor of #. Therefore A4,, the coefficient of 2" in the result ing series, will be equal to the sum of those coefficients a, whose suffix » is a divisor of # (including 1 or =). This we write sym- bolically pad Nay 2, dn and we then have, for |] ag > a, =X 4.7. 82 In words: the sum of all a,’s for which d is a divisor of =. 3 Any § 58. Special classes of series of analytic functions. — C. Lambert's series. 451 Examples. 1. a, =1. Here 4, is equal to the number of divisors of #n, which (as in 257, example 11) we denote by 7,; then PD zn 0 DT ET (lz1<1) n=1 n=1 =24+224 284834 +25 4148 4-27 445 4... In this curious power series, the terms z” whose exponents are prime numbers are distinguished by the coefficient 2. It was due to the misleadingly close connection between this special Lambert series and the problem of primes that this series (as a rule called simply tie Lambert series)®® played a consider- able part in the earlier attempts to deal with this problem. But nothing of importance was obtained in this manner. 2. a,=mn. Here 4, is equal to the sum of all the divisors of », which we will denote by z,’. Thus for [7] <1 Solan Zw Pr BRAS TALEA L120. SE 8. The relation 4, = 3 a, is uniquely reversible, i. e. for given 4,'s, the d/n coefficients a, can be determined in one and only one way so as to satisfy the relation. We then have in fact we Zo) se where wu (k) denotes the Mobius coefficients defined in 257, example 10, whose values are 0, +1 and —1. In consequence of this fact, not only can a Lambert series always be expanded in a power series, but conversely every power series may be expressed as a Lambert series, provided it vanishes for z=0,1.e. 4,=0. 4. For instance, if 4, =1 and every other 4, =0, en = 0 1) and we have the curious identity @D zn 2= Fr O77 (lz] <1). n=1" 5. Similarly, we find the representation, valid for |z| <1, 0 zn - = 1) ee d= 2 27¢ T= where ¢ (n) denotes the number of integers less than # and prime to #, — a number introduced by Euler. 6. Writing 2 ay, —— = =f(? and > a, 7" =g (4), and grouping the terms by Freon: in the double expansion as Lambert series on p. 450 (which is allowed), we obtain FO=s@+e@ +: = 5 eG). 1 a n—1 9. For a, =(=1V "4 =u, =(-1)""19, sve) =n =. We 3 Lambert, J.: Anlage zur Architektonik. Vol. 2, p. 507. Riga 1771. 4592 Chapter XII. Series of complex terms. obtain in this way, successively, the following remarkable identities, valid for |2]| <1, in which the summations are taken from n=1 to 00: Dy Jen = 30 Yi Began Bl, RT. nme e) SO fe np Ter = 3 (le) <1) ete. 8. In the two identities d) and €) we have on the right hand side a series of logarithms (for which of course we take the principal values); thus simple connections can be established between certain Lambert series and infinite products. E. g. from the two identities in question: I —2")=¢%, with we— 3+ St (=1Hr-t zn n Tn? AQ +2") =e", wih w=) 9. As an interesting numerical example we may mention the following: Taking #,=0, u, = 1 and for every #>1, 4, =up—1-+ up—2, we obtain Fibonacci’s sequence (cf. 6, 7) 0, 1, 4,82; 8, .5,. 8, 15: 2s. 94:55, We then have FL =i lelol ls ew where L (z) denotes the sum of the Lambert series >) ~ 8. The proof is based Te on the fact, which is easily established, that ip Ls hl (=0,%1,2,...) where « and § are the roots of the quadratic equation 2? —2—1=0. (Cf. Ex, 114) Exercises on Chapter XII®. 174. Suppose z,—( and b,—>b=+0. Under what conditions may we infer that b,%* — b%? 8 Landau, E.: Bull. de la Soc. math. de France, Vol. 27, p. 298. 1899. % In these exercises, wherever the contrary does not follow clearly from the context, all numbers are to be regarded as complex. Exercises on Chapter XII. 453 175. Suppose z;— 00 (i. e. | z,|— 4 00). Under what conditions may we then infer that a) (1 +2) —e?, Zn 1 b) ir = % — log 2? 176. The principal value of z' remains bounded for all values of z. 177. If f= Se {201 either z,— 0 or a 0, according as R (2) > 0 or << 0. What is the behaviour of (z,) when R (2) = 0? 178. Let a, b, ¢, d be four constants for which ad —bc==0 and let z, be arbitrary. Investigate the sequence of numbers (2), 2,, %,...) given by the recurring formula az,-+b rie : 1 What are the necessary and sufficient conditions that (z,) or (5) should converge? n - 2p+1 = w=0, 1,9, 5. And if neither of the two converges, under what conditions can z, become =z, again for some index p? When are all the z,’s identically equal? 179. Let a be given = 0 and z, chosen arbitrarily, and write for each n=0 1 a tues =5 (aro . (z,) converges if, and only if, z, does not lie on the perpendicular to the straight line joining the two values of Va through its middle point. If this condition is fulfilled, (z,) converges to the value of \/a nearest to z,. What is the behaviour of (z,) when z, lies on the perpendicular in question? ; 1 . 180. The series rs does mot converge for any real «; the series n 1 nl log nm 2 , on the other hand, does converge for every real o + 0. 181. For a fixed value of z and a suitable determination of the logarithm, does 1 1 Bs dee tip — log (s-+1)| tend to a limit as #» — + 00? 182. For every fixed z with 0 —— PR a =0; Z% Oo Nala c EE eh fr | ie Sie 9 3 186. If 3 a,z" converges for [2] <1 and its sum is numerically <1 for all such values of z, then X'|a, |? converges and its sum is <1. = 7 , where g, has the same meaning as in Ex. 47. n log 187. The power series Zn z2k-1 Byala JA ey S2k—1 TU=liietgery: Gr T-lride 1 z” A cf B{- YS rhe I fi =D ail 1 zt 227 g) Zon ne 2n—1)-2n : all have the unit circle as circle of convergence. On the circumference, they also converge in general, i. e. with the possible exception of isolated points. Try to express their sums by means of closed expressions involving elementary functions; separate the real and imaginary parts by writing z= 7 (cos z+ sin z), and write down the trigonometrical expansions so obtained for » << 1 and for r=1 separately. For which values of z do they converge? What are their sums? Are they the Fourier series of their sums? 188. What are the sums of the following series: COS NX COS MY cosnxsinny a B) SE n n sin na sin ny a | and of the three further series obtained by giving the terms of the above series | the sign (—1)"? Exercises on Chapter XII. 455 189. Proceeding with the geometric series 2 2” as in Ex. 187, but leaving r <1, we obtain the expressions 1—7cosa oo a 1 COSND = mm ) 1—2vcosx+ 72’ n=0 y sin x n 1 —_— by SL tauns 1—2rcosz}r®’ n=1 Deduce from them the further expansions S Cosnx c) 2 Tews a)™ [cos a)” =cos2z, sinnx d) 2 Teo Een =sin2% and indicate the exact intervals of validity. 190. In Exercise 187a the following expansion will have been obtained, among others: Rip 1 y sin x > ——sinnx=tan— “am 1—7rcosx Deduce from it the expansions 00 a) M(—D*1rrsin"g-sinna=tan—1(r4 cot x) (% — a) ’ n=1 2 cos®zsinnxy mw b) = SE TRL Ook and determine the exact intervals of validity. 191. Determine the exact regions of convergence of the following series: (1) 1 a) ; RE nr 9X 9 3-5 yu ! bal 2 1 1 > zn—1 DR Yr) f) 3(L 2), 2) (4 Zyeaven, where (p,) is real and increases monotonely to + oO. 192. Establish the relations an n oY 272. 142 a) TT = a zn ; n—1 by 3 ie= prt Sits where the summation begins with n=1. 456 Chapter XII. Series of complex terms. 193. Corresponding to Landaw’s theorem (271) we have the following: The Dirichlet series > (—1)"—1! a and the so-called binomial coefficient series n i z—1 5 ; a, 2 are in every case convergent and divergent together. 194. For which values of z does the equation Derr =e n=0 7 hold good? 195. Determine the exact regions of convergence of the following in- finite products: 9 71-5). Bn a(-4, oy JIA +#£21Y, d) I A+n2em); Slot 0 120-27) [ 5 Zi3(E) Tey +E) g) ala-2)& on 7 on | 5 aren h) n(-%), i) n(i-5rs), k) (1-524). \ 196. Determine, by means of the sine product, the values of the products . x? x4 0 ) M(1+5), »H(1+%), © (+5) for real values of x. The second of these has the value Se [cosh (zx V2) —cos (zx V2] : Does this continue to hold for complex values of x? 197. The values of the products 195, 1) and k), canbe determined in the form of a closed expression by means of the I'-function. 198. For |z]| <1, Aaa — +90 + a+). 199. By means of the sine product and the expansion of the cotangent in partial tractions, the following series and product may be evaluated in the + ® form of closed expressions; x and y are real, and the symbol 3 f(») indicates n=—ow + ® +» the sum of the two series 3 f(n) and 3 f(—%), and similarly for the n=0 k=1 product: + ® 1 +» 1 a) 2 GTA b) 2 nit xt’ +x 1 +o 4 x2 Lobel d to) I 2 Ew a Gta) § 59. General remarks on divergent sequences, 457 Chapter XIII Divergent series. § 59. General remarks on divergent sequences and the processes of limitation. The conception of the nature of infinite sequences which we have set forth in all the preceding pages, and especially in §§ 8—11, is of comparatively recent date; for a strict and irreproachable construction of the theory could not be attempted until the concept of the real number had been made clear. But even if this concept and any one general convergence test for sequences of numbers, say our second main criterion, were recognized without proof as practically axiomatic, it nevertheless remains true that the theory of the convergence of infinite sequences, and of infinite series in particular, is far more recent than the extensive use of these sequences and series, and the discovery of the most elegant results of the subject, e. g. by Euler and his con- temporaries, or even earlier, by Leibniz, Newton and their contem- poraries. To these mathematicians, infinite series appeared in a very natural way as the result of calculation, and forced themselves into notice, so to speak: e.g. the geometric series 1+ +4 a2 }-.. oc-. curred as the non-terminating result of the division 1/(1 — @); Taylor's series, and with it almost all the series of Chapter VI, resulted from the principle of equating coefficients or from geometrical considerations. It was in a similar manner that infinite products, continued fractions and all other approximation processes occurred. In our exposition, the symbol for infinite sequences was created and then worked with; it was not so originally, these sequences were there, and the question was, what could be done with them. On this account, problems of convergence in the modern sense were at first remote from the minds of these mathematicians®. Thus it is not to be wondered at that Euler, for instance, uses the geometric series 1 2 see —— 1+ 2+ 224 To even for x = — 1 or £ = — 2, so that he unhesitatingly writes ttt l tte Ls 1 Cf. the remarks at the beginning of § 41. 2 This relation is used by James Bernoulli (Posit. arithm., Part 8, Basle 1696) and is referred to by him as a “paradoxon non inelegans”. For details of the violent dispute which arose in this connection, see the work of R. Reiff men- tioned in 69, 8. 458 Chapter XIII. Divergent series. or 1 1-249 — 284 — =o; 3 2 similarly from (=) =1-}+2x +} 32% --- he deduces the relation 1 123 —d trim and a great deal more. It is true that most mathematicians of those times held themselves aloof from such results in instinctive mistrust, and recognized only those which are true in the present-day sense. But they had no clear insight into the reasons why one type of result should be admitted, and not the other. Here we have no space to enter into the very instructive dis- cussions on this point among the mathematicians of the 17% and 18th centuries®. We must be content with stating, e. g. as regards in- finite series, that Euler always let these stand when they occurred naturally by expanding an analytical expression which itself possessed a definite value®. This value was then in every case regarded as the sum of the series. : It is clear that this convention has no precise basis. Even though, for instance, the series1 —1-+1— 1-4 — -.- results in a very simple manner from the division 1/(1 — «) for # = — 1 (see above), and there- fore should be equated to there is no reason why the same series 5 should not result from quite different analytical expressions and why, in view of these other methods of deducing it, it should not be given a different value. The above series may actually be obtained, for x =0, from the function f(x) represented for every x > 0 by the Dirichlet series ® (— nt 1osgily piieg =2F-Tlontempthin or from 14x i—z rg oo Pt =a a! al f= te putting ® = 1. In view of this latter method of deduction, we should 2 : : have to take 1 — 1 --- == and in the case of the former there is no immediate evidence what value £(0) may have; it need not at any 1 rate be +5 3 For details, see R. Reiff, loc. cit. 1 In a letter to Goldbach (7. VIII. 1745) he definitely says: ... so habe ich diese neue Definition der Summe einer jeglichen seriei gegeben: Summa cujusque seriei est valor expressionis illius finitae, ex cujus evolutione illa series oritur’. § 59. General remarks on divergent sequences. 459 Euler's principle is therefore insecure in any case, and it was only Euler's unusual instinct for what is mathematically correct which in general saved him from false conclusions in spite of the copious use which he made of divergent series of this type®. Cauchy and Abel were the first to make the concept of convergence clear, and to renounce the use of any non-convergent series; Cauchy in his Analyse algébrique (1821), and Abel in his paper on the binomial series (1826), which is expressly based on Cauchy's treatise. At first both hesitated to take this decisive step®, but finally resolved to do so, as it seemed unavoidable if their reasoning were to be made strict and free from gaps. We are now in a position to survey the problem from above, as it were; and the matter at once becomes clear when we remember that the symbol for an infinite sequence of numbers — in whatever form it is given, sequence, series, product or otherwise — has, and can have, no meaning whatever in itself, but that a meaning was only assigned to it by us, by an arbitrary convention. This convention consisted firstly in allowing only convergent sequences, i. e. sequences whose terms approached a definite and unique number in an absolutely de- finite sense; secondly, it consisted in associating this number with the infinite sequence, as its value, or in regarding the sequence as no more than another symbol (cf 41,1) for the number. However ob- vious and natural this definition may be, and however closely it may be connected with the way in which sequences occur (e. g. as suc- cessive approximations to a result which cannot be obtained directly), a definition of this kind must nevertheless in all circumstances be con- sidered as an arbitrary one, and it might even be replaced by quite different definitions. Suitability and success are the only factors which can determine whether one or the other definition is to be preferred; in the nature of the thing itself, that is to say, in the symbol (s,) of an infinite sequence ?, there is nothing which necessitates any preference. We are therefore quite justified in asking whether the compli- cation which our theory exhibits (in parts at least) may not be due 5 Cf. on the other hand p. 133, footnote 6. 6 So far as Cauchy is concerned, cf. the preface to his Analyse algébrique, in which, among other things, he says: Je me suis vu forcé d’admettre plu- sieurs propositions qui paraitront peut-étre un peu dures, par exemple qu'une série divergente n’a pas de somme. As regards Abel, cf. his letter to Holmboe (16. I. 1826), in which he says: Les séries divergentes sont, en général, quelque chose de bien fatal, et c’est une honte qu'on ose y fonder une démonstration. — Moreover, J. d'Alembert had expressed himself in a similar sense as early as 1768. 7? (s,) may be assumed to be any given sequence of numbers, in particular, therefore, the partial sums of an infinite series Xa, or the partial products of an infinite product. We use the letter s because infinite series are by far the most important means of defining sequences. 261. 262. 460 Chapter XIII. Divergent series. to our interpretation of the symbol (s,)s as the limit of the sequence, assumed convergent, being an unfavourable one, — however obvious and ready-to-hand it may appear. Other conventions might be drawn up in all sorts of ways, among which more suitable ones might per- haps be found. From this point of view, the general problem which presents itself is as follows: A particular sequence (s,) is defined in some way, either by direct indication of the terms, or by a series or product, or otherwise. Is it possible to associate a “value” s with it, in a reasonable way? “In a reasonable way” might perhaps be taken to mean that the number s is obtained by a process closely connected with the previous concept of convergence, that is to say, with the formation of lims, = s. This has been found so extraordinarily efficacious in all the preceding that we will not depart from it to any considerable extent without good reasons. “In a reasonable way” might also, on the other hand, be inter- preted as meaning that the sequence (s,) is to have such a value s associated with it that wherever this sequence may occur as the final result of a calculation, this final result shall always, or at least usually, be put equal to s. 2 Let us first illustrate these general statements by an example. The series : S(—1)'=1—1+1—1+ —---, i. e. the geometric series 2a" for * = — 1, or the sequence {s.)=10,10, 1,0,:..; has so far been rejected as divergent, because its terms s, do not approach a single definite number. On the contrary, they oscillate unceasingly between 1 and 0. This very fact, however, suggests the idea of forming the arithmetic means Sot Siteed Smt 2 (n=0, 1, 2, io) . 1 n Since s, = 5 [1+ (—1)"], we find that 1 n n= 2m+1) =e Tey’ so that 5s,’ (in the former sense) approaches the value 3: : 1 lim 5, = - By this very obvious process of taking the arithmetic mean, we have accordingly managed, in a perfectly accurate way, to give a meaning to Euler's paradoxical equation 1 —1+4+1— +--+ =, to § 59. General remarks on divergent sequences. 461 associate with the series on the left hand side the number ' as its “value”, or to obtain this number from the series. Whether we can always equate the final result of a calculation to 5 whenever it ap- pears in the form X(— 1)", cannot of course be determined off-hand. In the case of the expansion yoo. 22 for = — 1, itis certainly go Zl . . 50; in the case of py for 2 =0, it 1s equally true, as may be shown by fairly simple means (cf. Exercise 200); — and a great deal more evidence can be adduced to show that the association of : . 1 : ; the sequence 1, 0,1, 0, 1, ... with the value 7 obtained in the manner described above is “reasonable” 8, We might therefore, as an experiment, make the following de- finition. If, and only if, the numbers s end SoS +:+:1 5, n n1 tend to a limit s in the previous sense, the sequence (s ), or series =a, will be said to “converge” to the “limit”, or “sum”, s. The suitability of this new definition has already been demon- strated in connection with the series 2'(— 1)", which now becomes convergent “in the new sense”, with the sum Tyy == which seems thoroughly reasonable. Two further remarks will illustrate the ad- vantages of this new definition: 1. Every sequence (s ), convergent in the former sense and of limit s, is so constituted, in virtue of Cauchy's theorem 43, 2, that it would also have to be called convergent “in the new sense”, with the same limit s. The new definition would therefore enable us to accomplish at least all that we could do with the former, while the example of the series 2'(— 1)" shows that the new definition is more far-reaching than the old one. 2. If two series, convergent in the old sense, 2'q,=— A and 2b, = B, are multiplied together by Cauchy's rule, giving the series - 2¢,=2(a,b,+a,b,_,+ a,b), we know that this series is not necessarily convergent (in the old sense). And the question when 2c, does converge presents very considerable difficulties and has not been satisfactorily cleared up so far. The second proof of theorem 189, 3 8 From the series (see above) for a also we can accordingly de- 3 duce the value 2 for xz =1. We have only to observe that the series, written ~ somewhat more carefully, is 1+0.2— 2>4+ 2340.2 — 2% ++ —--+, and is therefore 14+-0—-14+14+0—-14++4—...for z=1. - 16 = 469 Chapter XIII. Divergent series. however, shows that in every case Btn if C, denotes the nth partial sum of X¢,. The meaning of this is that Zc always converges in the new semse, with the sum AB. Here the advantage of the new convention is obvious: A situation which, owing to the insuperable difficulties involved, it was impossible to clear up as long as we kept to the old concept of convergence, may be dealt with exhaustively in a very simple way, by introducing a slightly more general concept of convergence. We shall very soon become acquainted with other investigations of this kind (see § 61 in particular); first of all, however, we shall make some definitions relating to several fundamental matters: Besides the formation of the arithmetic mean, we shall become acquainted with quite a number of other processes, which may with success be substituted for the former concept of convergence, for the purpose of associating a number s with a sequence of numbers (s,). These processes have to be distinguished from one another by suitable designations. In so doing it is advisable to proceed as follows: The former concept of convergence was so natural, and has stood the test so well, that it ought to have a special name reserved for it. Accor- dingly, the expression: “convergence of an infinite sequence (series, product, ...)” shall continue to mean exactly what it did before. If by means of new rules, as, for instance, by the formation of the arithmetic mean described above, a number s is associated with a sequence (s,), we shall say that the sequence (s,) is limitable* by that process, and that the corresponding series X'a, is summable by the process, and we shall call s the value of either (or in the case of the series, its sum also). ; When, however, as will occur directly, we are making use of several processes of this kind, we distinguish these by attached initials 4,B, ...,V,..., and speak for instance of a V-process®. We shall say that the sequence (s,) is limitable V, and that the series Xa, is summ- able V; and the number s will be referred to as the V-limit of the sequence or V-sum of the series; symbolically V-lims,=s, V-2a,=s. When there is no fear of misunderstanding, we may also express the * German: limitierbar. 9 In the case of the concept of ¢nfegrability the situation is somewhat similar and it was probably in this connection that the above type of notation : was first introduced. Thus we say a function is infegrable R or integrable L according as we are referring to integrability in Riemann’s or in Lebesgue’s sense, § 59. General remarks on divergent sequences. 463 former of the two statements by the symbolism Vis )—%, which more precisely implies that the new sequence deduced from (s,) by the V-process converges to s. When, as will usually be the case in what follows, the process , includes several stages, we attach a suffix and speak of a V,-limitation process, a V,- summation process, etc. In the construction and choice of such processes we shall of course 263. not proceed quite arbitrarily, but we shall rather let ourselves be guided by questions of suitability. We must give the first place to the fundamental stipulation to be made in this connection, namely that the new definition must not contradict the old one; indeed, the latter has proved so eminently useful that a definition contradicting it could scarcely be expected to be of much use. We accordingly stipu- late that any TV-process which may be introduced must satisfy the following permanence condition: I. Every sequence (s,) convergent in the former sense, with the limit s, must be limitable V with the value s. Or in other words, ims, =s must in every case imply V-lim s, = s?°. ™ In order that the introduction of a process of this kind may not be superfluous, we further stipulate that the following extension con- dition is to hold: II. At least one sequence (s,), which diverges in the former sense, must be limitable by the new process. Let us call the totality of sequences which are limitable by a particular process the range of action of this process. The condition II implies that only those processes will be allowed which possess a wider range of action than the ordinary process of convergence. It is precisely the limitation of formerly divergent sequences and the summation of formerly divergent series which will naturally claim the greater part of our attention now. Finally, if several processes are employed together, say a V-process and a W-process simultaneously, we should be in danger of hopeless confusion if we did not also stipulate that the following compatibility condition should be fulfilled: IL If one and the same sequence (s,) is limitable by two different processes, simultaneously applied, then it must have the same value by both processes. In other words, we must in every case have V-lims, = W-lims_, if both these values exist. 10 We might also be satisfied if some convergent sequences at least are limitable with unaltered value by the process considered. This is the case e.g. with the E,-process discussed further on, provided the suffix p is complex. 264. 265. 464 Chapter XIII. Divergent series. We shall only consider processes which satisfy these three con- ditions. Besides these, however, we require some indication whether the association of a value s with the sequence (s,) effected by a parti- cular V-process is a reasonable one in the sense explained above (p- 460). Here widely-varying conditions may be laid down, and the processes which are in current use are of very varied degrees of ef- ficiency in this respect. In the first instance we should no doubt require that the elementary rules of the algebra of convergent sequences (v. § 8) should as far as possible be maintained, i. e. the rules for term-by-term addition and subtraction of two sequences, term-by-term addition of a constant, and term-by-term multiplication by a constant, and the effect of a finite number of alterations (27, 4), etc. Next we might perhaps require that if, say, a divergent series Xa _ has associated with it the value s, and if this series is deduced, e. g. from a power series f(x) = Zc,a" by substituting a special value z, for z, then the number s should bear an appropriate relation to f(x) or to lim f(z) for z —x,; and similarly for other types of series (Dirichlet series, Fourier series etc.). In short, we should require that wherever this series appears as the final result of a calculation, the result should be s. The greater the number of conditions similar to the above which are satisfied by a particular process — let us call them the conditions F, without taking pains to formulate them with absolute precision — and at the same time, the greater the range of action of the process, the greater will be its usefulness and value from our point of view. We proceed to indicate a few of these processes of limitation which have proved their worth in some way or another. 1. The Ci- or Hy-process!!, As described above, 262, we form the arithmetic means of the terms of a sequence (s,): So+S; +--+ 3 eR (n=0,1,2,..) which we will denote by ¢ or 7". If these tend to a limit s in the older sense, when #-—>00, we say that (s,) is limitable Cy or limitable Hy with the value s and we write Colims,=s or Cy(s,)—s or with H, instead of C,. The series Xq, with the partial sums s, will be called summable C; or summable Hy, and s will be called its C,- or H;-sum. The sequence of units 1,1,1,... may be considered to be the simplest convergent sequence we can conceive. The process described above consists in comparing on the average, the terms s,_ of the sequence 11 The choice of the letters C and H is explained in the two next sub- sections. § 59. General remarks on divergent sequences. 465 under consideration with those of the sequence of units: el niin = =% TT 14. This “averaged” comparison of (s,) with the unit sequence will be met with again in the case of the following processes. The usefulness of this process has already been illustrated above by several examples. We have also seen that it satisfies the two con- ditions 268, I and II, and III does not come under consideration at the moment. In §§ 60 and 61 it will further be seen that the con- ditions F (264) are also in wide measure fulfilled. 2. Holder's process, or the H -process!’. If with a given sequence (s,), we proceed from the arithmetic means 4,’ just formed to ther mean nh ". lth etl) % Ty n=0,1,2,.,3 and if the sequence (%,”) has a limit in the ordinary sense, lim /,”=s we say that the sequence s, is limitable Hy with the value s*2. By 43,2, every sequence which is limitable H, (and therefore also every convergent sequence), is also limitable H,, with the same value. The new process therefore satisfies the conditions 263, I, II and III; moreover, its range is wider than that of the H,-process, for the series J) PE) EL 2h mr n=0 ; : i for instance, is summable H, with the sum 1 but not summable H, nor convergent. In fact, we have here (s,) = 1 —1, 2. rm 2, 3, rons 3s coe and 2 3 > 5 0, 5? These sequences are not convergent. On the other hand, the numbers rH=1,0 B,0.50, b=, as is easily calculated. This is precisely the value which one would expect from LX 8 (=) = n+" n= o for x = —1. 1? Holder, O.: Grenzwerte von Reihen an der Konvergenzgrenze. Math. Ann., Vol. 20, pp. 535—549. 1882. Here arithmetic means of the kind described are for the first time introduced for a special purpose. 13 The rest of the notation is formed in the same way, H,-lim s,=s, Hy-Za,=s, H,(s,) —s, etc. but hereafter we shall not mention it specially. 466 Chapter XIII. Divergent series. If the numbers %,” do not tend to a unique limit, we proceed to take ther mean ” hr ele rr B= Ph (n=0, 1, 2, ves) or, in general, the mean PU r=. Sek 22-0 dy 22 vv) oT (n=0,]1 ) nP i between the numbers 52" if these new numbers WD on s, for some definite p, we say that the sequence (s,) is limitable H,, with the value s. ~ It is easy to form sequences which are limitable H_ for any particular given p, but for no smaller value of p than this!5. This, together with 43, 2, shows that the H, -processes not only satisfy the conditions 263, I—III, but that their range of action is wider for each fixed p > 2 than for all smaller values of 4. As regards the con- ditions F, we must again refer to §§ 60 and 61. obtained at the previous stage (p> 2); 8. Cesaro’s process, or the C,-process?®, We first write = S52 and also, Tor each 22> 1, gi, Ze) SE Li gD) oo) Sa, (n=0, 1: 2; ves) and we now examine the sequence of numbers (k) ._ 5 17 n ® 1 J 9 k for each fixed k. If, for some value of %, Ds, we say that the sequence (s,) is limitable Cj, with the value s. In the case of the H-process, we cannot obtain simple formulae giving nh? directly in terms of s,, for larger values of p. In the case of the C-process, this is easily done, for we have SO= (12a CET + (IT) 14 Or indeed for p=1, provided we agree to put WO = s,, as we shall do here and in all analogous cases in future. 15 Write, for instance, 20) =1,0,1,0,1,... and work backwards to the values of s,. Other examples will be found in the following sections. 16 Cesaro, E.: Sur la multiplication des séries. Bull. des sciences math. (2), Vol. 14, pp. 114—120. 1890. 1? The denominators of the right hand side are exactly the values of sh obtained by starting with the sequence (s,)=1,1,1,.... Thus the C-process ° again involves an “averaged” Sompanisey between a given sequence (s,) and the unit sequence. . § 59. General remarks on divergent sequences. 467 or if we wish to go back to the series 2a, with the partial sums s,, k E—1 k stm (Fat (Hak (De This may be proved quite easily by induction, or by noticing that, by 102, S'sy SG%- Dn = {1 —u) ISH" n=0 n=0 so that for every integral ; =0 1 a n = aT 2% » ool) on __ Seber — ty Baat=r whence, by 108, the truth of the statement follows 18, In the following sections we shall enter in detail into this process also, which becomes identical with the preceding one (bh, = ¢,") for p—1. 4. Abel's process, or the A-process. Given a series 2'g, with the partial sums s_, we consider the power series tay=2a 0" = ~n)3s, 2 If its radius is >1, and if (for real values of x) the limit Im Za a= lim 1—2) 2s, a"=3s z>1-0 z>1—-0 exists, we say that the series 2a, is summable A, and that the sequence (s,) is limitable A, with the value s°; in symbols: A-Za,=s, A-lims =g. In consequence of Abel's theorem 100, this process also fulfils the permanence condition I, and simple examples show that it fulfils the “extension condition” II; for instance, in the case of the series 2 (—1)" already used, the limit for £—1—0 lim (2 (—1)" 2") = lim = : exists. Thus Euler's paradoxical equation (p. 457) is again justified 18 Tn view of these last formulae, it is fairly natural to allow non-integral values > — 1 for the suffix £ also. Such limitation processes of non-integral order were first consistently introduced by the author (Grenzwerte von Reihen bei der Anndherung an die Konvergenzgrenze, Inaug.-Diss., Berlin 1907). We shall not enter into this question, either here in the case of the C-process, or later in that of the other processes considered. 19 Jf the product (1 —z) 2's, x" is written in the form 23,7 Ey we see that it is again an “averaged” comparison of the given sequence with the unit sequence which is involved, though in a somewhat different manner. 468 Chapter XIII. Divergent series. by this process. If we now use the more precise form AZS)'=5 or GoZ{=1=t, we thus indicate two perfectly definite processes by which the value dl may be obtained from the series X'(—1)". 5. Euler's process, or the E-process. We saw in 144 that if the first of the two series X( Ya, and S49 — n [= aktl converges, then so does the second, and to the same sum. Simple examples show, however, that the second series may quite well con- verge without the first one doing so: 1. lig =1, then g,=1 and 4%4,~0 for £21. Accordingly, the two series are 1—-14+1—-1-}—-.. and 1 5 +O0+0+0+ 1 the second of which converges to the sum —. 2 2. If, for n=0,1,2,..., 7) == 3; 2, 3, dines; then da, = —1, —1, —1, — 1, cee y and for k>2 4%a, = 0, 0: 0,>0,.... Accordingly, the two series are 1 1 1—24+3—4-+—-... and wg 004. the second of which converges to the sum 7 3. Similarly for a,=(n 11)? we find da, =—17, 4%a,=12, 4%a,=—6, and, for 2 >3, A4%a,— 0. The two series are thus 1 7 12 6 1—8-}-27 —64-F—... and TTT TT TOE 0A the second of which converges to the sum — —. 8 4. For a, = 2", A¥a,=(—1)* Thus the two series are: 1 1 1 1 1—2+4—8+}+—-.- and Gels ee, the second of which converges to the sum . — a value which we should expect from == Sa" {Or B13, 5 For g,=(—1)"7" da, SE (1-1-7 The two series are therefore x 2" and > Gy n=0 on .aku the second of which converges to the sum sk provided [241] < 2. i § 59. General remarks on divergent sequences. 469 — If we start with any series Sa, , without alternately 4 and — signs, the series > = . 1 n n nN Saf with a/=5((5) at (Dat + (5) al] will be an Euler's transformation of the given series, which we may also obtain as follows: The series Xa, results from the power series Za, x? for x =1, hence from ©» n+1 > a5) = - for y= Expanding the latter in powers of y, before substituting 1 . , : Y= 5, WE obtain Euler's transformation. In fact 2 a, wht! = > a, (5) = Se, > Sa yhtit1 = HP = E a} grit Bight in giesd, n=0 n=" In order to adapt this SE for use with any sequence (s,) we write, deviating somewhat from the usual notation, tyra ws for nl, ond 5,=0, and also a, +a’+ tag 4=s' for n2=1, and s/=0. It is now easy to verify that for every un > 0 orm de (St (oto (a) We accordingly make the following definition: A sequence (s,) is said to be limitable Ey; with the value s, if the sequence (s/) just de- fined tends to s 2. If, without testing the convergence of (s,’), we write * From Sa,a"t =a, 2)" it follows, by multiplication by 1 1—y =. t 1— = Frye w Er (1=9 3 ol @3)" k=0 Hence J on w ow 0 2 wey gh Za y y- 2 F(A qe n=0 1=y 1-9) ifn iah\i% k= - ZA [(3) er (ares (Delon whence the relation may at once be inferred. 21 Here also the denominator 2” is obtained from the numerator [(§) 0-3) =] by replacing each of the s,’s by 1. Thus we are again concerned with an “averaged” comparison, of a definite kind, between the sequence (s,) and the unit sequence. 16% 470 Chapter XIII. Divergent series. w= Se l(5)ss + (Far (2) and in general, for » > 1, 1 iv on (2 jo n\ r-1 $2 i (5) 1 + (7) Dp (Mar 1 {n=0,1; gi) we shall similarly say that the sequence (s,) is limitable E, and regard s as its EF -limit, if, for a particular 7, Pg Our former theorem 144 (see also 44, 8) then shows in any case that this E- process satisfies the permanence condition I, and the exampies given there show that the condition II is also satisfied. This process will be examined further in § 63. 6. Riesz’s process, or the R, -process?>. For making the principle of averaged comparison of the sequence (s,) with the unit sequence more powerful, — a principle which, as we saw, lies at the basis of all the former limitation processes, — a fairly obvious pro cedure consists in attributing arbitrary weights to the various terms s, . If 2,, 4, 45, ... denote any sequence of positive numbers, strictly mono- tone increasing and tending to -}- oo, and if we write b= =, dor pn 21 wd} =u, the numbers r__ HoSot Sit Fass §, =" tr 7 n are generalized mean values. As with the H-, C-, or E-processes, this generalized method of forming the mean may of course be repeated, writing, for instance, as in the C-process, S$, = a and in =1, and then, for 2 i; (k) ar -1) (1) (k-1) op = 0 7 Aol yp; 18 Ho d= Nin. J &E-1 nn ’ and then proceeding to investigate, for fixed £>1, the ratios and (%) o o® em n 2 (k) n for n — +00. If these tend to a limit s, we might say that (s,) was limitable R,, ** with the value s. This definition, however, is not in use. The process in question has reached its great importance only by being transformed into a form more readily amenable to analysis, as 22 Riesz, M.: Sur les séries de Dirichlet et les séries entiéres. Comptes rendus, Vol. 149, pp. 909—912. 1909. 23 Here we add a suffix 1 to Ry, the notation of the process, as a refe- rence to the sequence (4,) used in the formation of the mean. For i, =n +1 this process reduces exactly to the Ci-process. — § 59. General remarks on divergent sequences. 471 follows: A (complex) function s (£) of the real variable t > 0 is defined by s{=3s, In J_, +® lim aE (8) (0 — Hb-14t. 0 If this limit exists and =s, the sequence (s,) will be called limitable Ry with the value s. Here we cannot enter into a more detailed examination of the question whether the two definitions given for the R,,-process are really exactly equivalent, or into the elegant and far-reaching appli- cations of the process in the theory of Dirichlet's series. (For refer- ences to the literature, see p. 477.) 7. Borel’s process, or the B-process. We have just seen how Riesz’ process tends to increase the efficiency of the H- or C-pro- cesses, by substituting for the method of averaged comparison be- tween the sequence (s,) and the unit sequence a more general form of this procedure. The range of Abel's process may be enlarged in a similar way by making use of other series instead of the geometric series there used for purposes of comparison. Taking the exponential series as a particular case, and accordingly considering the quotient of the two series PD xh @0 xt Seo md Fo, n=0 ! n=0 24 The equality of the two sides is easily proved by induction, using inte- gration by parts. 4792 Chapter XIII. Divergent series. that is to say, the product = ee x» Fli=g% 28,7 n=0 for x — + co, we obtain the process introduced by E. Borel?®. In accordance with it we make the following definition: A sequence (s,) n = such that the power series 3's ey converges everywhere and the function F(x) just defined tends to a unique limit s as x— 4 co, will be called limitable B with the value s. In order to illustrate the process to some extent, let us first take 2a, = 2(— 1)" once more; then s, =1 or 0, according as # is even or odd. Se Xs Sn H =1 + 21 +o 4! Le 2 and we have to deal with the limit ime" ge T>+ © 2 . which is evidently 2 Thus X(— 1)" is summable B with the sum > More generally, taking Ja = Xz", we have, provided only that z 41-1, 1-—27+1 ge and F@)=¢2- 3s 1 x ele 7 = 1—2z 1—2z which — a when — -}- co, provided R(z) <1. Thus the geometric series 2 2" is summable B with the sum = throughout the half- plane H(z) < 1%. This process also satisfies the permanence condition; for we have zn , ne —z, —_— — = eg %. ( —_— cs (e 8, a 020 3 (8. 8) If s,—»s in the ordinary sense, we can for any given choose m so 25 Sur la sommation des séries divergentes, Comptes rendus, Vol. 121, p. 1125. 1895, — and in many Notes in connection with it. A connected ac- count is given in his Legoas sur les séries divergentes, Paris 1901. ’ 26 By the C-processes, the geometric series is summable, beyond |z| <1, only for the boundary points of the unit circle, 41 excepted; by Euler's pro- cess it is summable throughout the circle |z-1| <2, which encloses the unit circle, with a wide margin; by Borel's process it is summable in the whole half-plane $i (z) <1, — the value in this and the preceding cases being every- where % i § 59. General remarks on divergent sequences. 473 large that |s, — s]| <3 for every n> m. The expression on the right hand side is then in absolute value So Sle sl Lge 3s, +2 —s|-Z nl for positive x's. Now the product of ¢~” and a polynomial of the mh degree tends to 0 when x— oo; we can therefore choose & so large that this product is < 5 for every x > £. For these #’s the whole expression is then < ¢ in absolute value and our statement is estab- lished. (For further references, see pp. 477—478.) 8. The B,-process. The range of the process just described n may now be further extended by substituting other series for J] nid n!’ : gh : ; in the first instance 2 ear where 7 is some fixed integer >1. We accordingly say that a sequence (s,) is limitable B, with the value s if the quotient of the two functions - rm rn = x a Zotar wd 2a n=0 that is to say, the product a xz’? — i re “3s Se lrm)l? tends to the limit s when £— 4- occ. (We must, of course, assume again here that the first-named series is everywhere convergent.) 9. Le Roy’s process. We have usually interpreted the limitation processes by saying that by means of them we carry out an “averaged” comparison between the given sequence (s ) and the unit sequence 1,1,1,... We may look at the matter in a slighdy different way. If the numbers s, are the partial sums of the series 2'a,, we have to examine, for instance in the C,- process, the limit of S85 ++, n+ 1 —at (t= ok (=a bo i= Here the terms of the series appear multiplied by wvariable factors which reduce the given series to a finite sum, or at any rate to a series convergent in the old sense. By means of these factors, the influence of distant terms is destroyed or diminished; yet as » in- creases all the factors tend to 1 and thus ultimately involve all the terms to their full extent. In the case of Abel's process, where we were concerned with the limit of Yq a" for —1 — 0, the effect 474 Chapter XIII. Divergent series. described above is brought about by the factors x". This principle appears most clearly as the basis of the following process®’: The + serjes $Leetl) n en n! is assumed convergent for 0 x, and that the series X¢, (x)s, converges for each of these values of xz. In that case F(x) is also defined for every « > x, and we may investigate the existence of the limit lim F(z). T>+ » If the limit exists and =s, the sequence (s,) will be called limitable ¢ with the value s3 By analogy with 221,2, we shall at once be able to assign con- ditions under which a process of this type will satisfy the permanence condition. This will certainly be the case if a) for every fixed n, lim Pa (©) =0, Tr if b) a constant K exists such tt [Po @)| + @)| + +9. @)| < K for every x > x, and all »’s, and if c) for x — } © lim (Xe, (x) =1. It will be noticed that these conditions correspond exactly to the as- sumptions a), b) and c) of theorem 221,2%. The proof, which is quite analogous to that of this theorem, may therefore be left to the reader. Borel’s process evidently belongs to this type, with ¢, (z) = eZ The same may be said of Abel's process, if the interval 0... co 8 The importance of theorem 221, 2 lies chiefly in the fact that the conditions a), b) and c) of the theorem are not merely sufficient, but actually necessary for its general validity. We cannot enter into the question v.p 72, footnote 19), but we may observe that in consequence of this fact, the T-processes whose matrix satisfies the conditions mentioned are the only ones which fulfil the permanence condition. 3 In all essentials this is the scheme by means of which O. Perron (Bei- triage zur Theorie der divergenten Reihen, Math. Zschr. Vol. 6, pp. 286—310. 1920) classifies all the summation processes. 32 Like these they are not only sufficient, but also necessary for the general validity of the theorem. ’ / 476 Chapter XIII. Divergent series. is projected into the interval 0...1 which is used in the latter, that is, if the series (1 — x) 3's, @" is replaced by the series 1 2 2 N Pre ire) and the latter is examined for #— 400. — In an equally simple manner, it may be seen that Le Roy's process belongs to this type. The second type of limitation process contains the first as a par- ticular case, obtained when «x assumes integral values > 0 only (p, (k) =a,,)- We merely use a continuous parameter in the one case, and a discontinuous one in the other. Conversely, in view of § 19, def. 4a, the continuous: passage to the limit may be replaced by a discontinuous one, and hence the ¢-processes may be exhibited as a sub-class of the T-processes. These remarks, however, are of little use: in further methods of investigation the two types of process nevertheless remain essentially different. It is not our intention to investigate all the processes which come under these two headings from the general points of view indicated above. Let us make only the following remarks: We have already pointed out what conditions the matrix T or sequence of functions (¢, ) must fulfil, in order that the limitation process based on it may satisfy the permanence condition 268, I. Whether the conditions 268, II and IIT are also fulfilled, will depend on further hypotheses regarding the matrix T or sequence (¢,); this question is accordingly best left to a separate investigation in each case. The question as to the ex- tent to which the conditions F (264) are fulfilled, cannot be attacked in a general way either, but must be specially examined for each process. One important property alone is common to all the 7- and g-processes, namely their linear character: If two sequences (s,) and (¢) are limitable in accordance with one and the same process, the first with the value s, and the second with the value #, then the sequence (as, + bt), whatever the constants ¢ and b may be, is also limitable by the same process, with the value as--b¢ The proof follows immediately from the way in which the process is constructed. Owing to this theorem, all the simplest rules of the algebra of con- vergent sequences (term-by-term addition of a constant, term-by-term multiplication by a constant, term-byterm addition or subtraction of two sequences) remain formally unaltered. On the other hand, we must expressly emphasize the fact that the theorem on the influence of a finite number of alterations (42, 7) does not necessarily remain valid 33. 33 For this, the following simple example was first given by G. H. Havdy, relating to the B-process: Let s, be defined by the expansion 3 , © xr - sinfe”)= Ys, aT n=0 2 Since e—7.sin (¢?) > 0 as x — 4 00, the sequence Sy) S35 85, ++. is limitable B § § 59. General remarks on divergent sequences. 477 If we wished to give a general and fairly complete survey of the present state of the theory of divergent series, we should now be obliged to enter into a more detailed investigation of the processes which we have described. To begin with, we should have to deal with the questions whether, and to what extent, the individual processes do actually satisfy the stipulations 263, II, III and 264; we should have to obtain necessary and sufficient conditions for a series to be summable by a particular process; we should have to find the relations between the ways in which the various processes act, and go further into the questions indicated in No. 10, etc. Owing to lack of space it is of course out of the question to investigate all this in detail. We must be content with examining a few of the processes more particulary; — we choose the H-, C-, A-, and E-processes. At the same time we will so arrange the choice of subjects that as far as possible all questions and all methods of proof which play a part in the com- plete theory may at least be indicated. For the rest we must refer to the original papers, of which we may men- 266. tion the following, in addition to those mentioned in the footnotes of this section and of the following sections: 1. The following give a general survey of the group of problems: Borel, E.: Legons sur les séries divergentes. Paris 1901. Bromwich, T. J. I’A.: An introduction to the theory of infinite series. London 1908: 2d ed. 1926. : Hardy, G. H., and S. Chapman: A general view of the theory of summable series. Quarterly Journal Vol. 42, p. 181. 1911. Chapman, S.: On the general theory of summability, with applications to Fourier’s and other series. Ibid., Vol. 43, p. 1. 1911. . Carmichael, R. D.: General aspects of the theory of summable series. Bull. of the American Math. Soc. Vol. 25, pp. 97—131. 1919. : Knopp, K.: Neuere Untersuchungen in der Theorie der divergenten Reihen. Jahresber. d. Deutschen Math.-Ver. Vol. 32, pp. 43—67. 1923. 2. A more detailed account of the Rj; j-process, which is not specially considered in the following sections, is given by Hardy, G. H., and M. Riesz: The general theory of Dirichlet’s series. Cambridge 1915. The B-process is dealt with in the books by Borel and Bromwich mentioned under 1., and also in more detail by Hardy, G. H.: The application to Dirichlet’s series of Borel’s exponential method of summation. Proceedings of the Lond. Math. Soc. (2) Vol. 8, pp. 301 to 320. 1909. with the value 0. By differentiation of the relation above, we obtain a x? cosi(e®)=e—7. 2 Sutin n= this shows, since cos (¢?) tends to no limit when z— + 00, that the sequence $1) Sas S3y - »» iS mot limitable B at all! 478 Chapter XIII. Divergent series. - R67. Hardy, G. H., and J. E. Littlewood: The relations between Borel's and Cesaro’s methods of summation. Ibid., (2) Vol. 11, pp. 1—16. 1913. Hardy, G. H., and J. E. Littlewood: Contributions to the arithmetic theory of series. Ibid., (2) Vol. 11, pp. 411—478. 1913. Hardy, G. H., and J. E. Littlewood: Theorems concerning the summability of series by Borel's exponential method. Rend. del Circolo Mat. di Palermo Vol. 41, pp. 36—53. 1916. Doetsch, G.: Eine neue Verallgemeinerung der Borelschen Summabilitéts- theorie. Inaug.-Diss., Gottingen 1920. 3. Apart from the books mentioned under 1., a full account of the theory of divergent series is to be found in Bieberbach, L.: Neuere Untersuchungen iiber Funktionen von komplexen Variablen. Enzyklop. d. math. Wissensch. Vol. II, Part B, No. 4. 1921. 4. Finally, the general question of the classification of limitation processes is dealt with in the following papers: Perron, O.: Beitrag zur Theorie der divergenten Reihen. Math. Zeitschr. Vol. 6, pp. 286—310. 1920. Hausdorff, F.: Summationsmethoden und Momentenfolgen I und II. Math. Zeitschr. Vol. 9, p. 74 seqq- and p. 280 seqq. 1920. § 60. The C- and H-processes. Of all the summation processes briefly sketched in the preceding section, the C- and H-processes — and especially the process of limitation by arithmetic means of the first order, which is the same in both — are distinguished by their great simplicity; they have, more- over, proved of great importance in the most diverse applications. We shall accordingly first examine these processes in somewhat greater detail. In the case of the H-process, Cauchy's theorem 43,2 shows that, for p> 1, A?" —s implies 4" — 53% so that the range of the H -process contains that of the H p_1-Process. The corresponding fact holds in the case of the C-process: , Theorem 1. If a sequence is limitable C,_, with the value s, (k=>1), it is also limitable C, with the same value. In symbols: From cP —»s, it follows that oe 5. (Permanence theorem for the C- process). 2 By £0 we mean simply s, itself, and similarly in the case of the other processes. Thus by the (Oth degree of a transformation, higher degrees of which are introduced, we mean the original sequence. . : § 60. The C- and H -processes. . 479 Proof. By definition (v. 2635, 3) ®) Se SPU... gl=0) Cn = = 2 Crp Piel wind fpf ial gel) E2—1 E—1 - 1 W-L— | ? LiL £1 ) whence by 44, 2 the statement immediately follows. Accordingly, to every sequence which is limitable C _, for some suitable suffix p, there corresponds a definite integer k such that the sequence is limitable C, but is not limitable C,_,. (If the sequence is convergent from the first, we of course take k= 0.) We then say that the sequence is exactly limitable C,. Examples of the Ci-limitation Process 3. 1. 3 (—1)* is summable C, with the value 3; Proof above, 262. n=0 2 ~~ ntk\ 1 2. oF 2p B ) is exactly summable Cg4i1, to the value s= SEF’ , we have by 265,3 Zeal) Tel Pe hE 1 v1 vs-8Y 9, -(1i%) re Accordingly : sP=(’ 1") or =0, according as n=2y» or =2»-}1. In fact, for a,=(—1)" (9 Hence both for n =2v and for n =2v-+1, G+ k ba C19 ih whence the statement follows immediately. : 3. The series J (— 1)" (n+ 1) =1—2%4 3% — 4% —... summable C, to the value ° for £=0, by Example 1., is for each k= 1 exactly summable C, , , 2k+1_ 1 : to the sum sey if B, denotes the wth of Bernoulli’s numbers. The fact of the summability indeed follows directly from Example 2. For the moment denoting the series there summed by 3}, we at once see, from the linear character of our process (v.p. 476), that the series, obtained from Xj 3 As a result of the equivalence theorem established immediately below these examples hold unaltered for the Hj-limitation processes. On account of the explicit formulae for Si and 22, given in 265,38, to which there is no analogue in the H-process, the C-process is usually preferred. 268. 480 Chapter XIII Divergent series. by term-by-term addition, of the form Coot 2+ +k 2k is exactly summable Cg+1 if cy, ¢y, ..., cx denote any constants, with ¢;,=0. — The value s is most easily obtained by A-summation; see 288, 1. 4. The series oe cos x -- cos 2x4 cCOSRTA- -+ is summable C, to the sum 0, provided x3: 2% a. Proof. By 201, 1 sin (n+3) @ Sp ry F008 Bek COS Wolo er CORNY Emin 2 sin for each #=0, 1, 2,...; heace ha z 1 oi am . i sin (+1) spb syt orbs =——— (sing sin85 4 psin@nt 1) ES 2 sin 5: 2 sin? and consequently SoS kre, 1 n-+1 5 1 n-+1 2 sin? - For a fixed x4 2k x, the expression on the right tends to 0 as » increases. which proves what was stated. — This is our first example of a summable series with variable terms. The function represented by its “sum” =0 in every interval not containing any of the points 2% x. At the excluded points, the series is definitely divergent to 4-00! 5. The series sinz--sin2z-+sin3 x4... is obviously convergent with the sum 0, for x =k x. For x= k= it is no longer convergent, but it is sum- 1 x 36 mable C,, and it then has the “sum” ~- cot 5 2 Proof. From the relation 1 cos(2n+1) = s,=sinz+...4sinnr=—_cot-———— ’ 2 2 2 sin = 2 the statement follows as in 4. 6. cosz-+cos3x+cosbaw+--- is summable C, to the sum 0, for z=Xia. 7. sin ¢ + sin 8x }sindx +... is also summable C; to the sum Tone’ for ztka. 8 1+4+2z1224... is summable C, on the circumference |z|=1, ex- cepting only for z=-41, and the sum is i ! (Examples 4 and 5 result from this by separating real and imaginary parts.) Here, in fact, SE soiinm BI utI ie, le on HI 1—2 1-2 n+ 1 1-2 n4+1 (1-22 whence the statement can be inferred at a glance. Sa 36 The graph of this function thus exhibits “infinitely great jumps” at the points 2k =. § 60. The C- and H-processes. x 481 9. The series Lh = > WL 2" remains summable Cj to the ra la) EN Ber] su on the circumference |z|=1, provided only z+ 1. For the 0 1 a—2) corresponding quantities sh are, by 2695,3, the coefficients of 2” in the expansion of 1 1 a A=) +1 [T—azzk (I—az) +1 dees, (the right hand side being the expansion in partial fractions of the left hand side). All the partial fractions after the one written down contain in the de- nominator the kth power of (1 —2x) or (1 —=x2) at most. Hence, multiplying by (1 —z)*+1 and letting x — 1, we at once obtain a a Accordingly © Sr 1 uti $ SM gn F ( Jee] =o n=0 (1 =nEN if : where it is sufficient to know that the supplementary terms within the square bracket involve binomial coefficients of the order n*—1! with respect to % at most. Therefore, as #— 4 00, ee q-e. d. (1H (1—2)% k Since the H-process outwardly seems to bear a certain relation- ship to the C-process, it is natural to ask whether their effects are distinguishable or not. We shall see that the two ranges of action coincide completely. Indeed we have the following theorem, due to the author?’ and to W. Schnee®S: Theorem 2. If a sequence (s,) is, for some particular k, sum- mable H, to the value s, it is also summable C, to the same value 269. s and conversely. In symbols: PE s always involves on 5s and conversely. (Equivalence theorem for the C- and H -processes.) i For the proof, we require a lemma, which for purposes of sub- sequent use we will at once formulate in a form which is somewhat more general than that required for our present purpose. Lemma. If for a given sequence (s,) the relation Ge Ape ofp So holds for some definite integral k > 0 and some definite integral p => 0, 37 Cf. the paper cited on p. 467, footnote 18. 38 Schnee, W.: Die Identitdt des Cesaroschen und Hoilderschen Grenzwertes. Math. Ann. Vol. 67, pp. 110—125. 1909. 3 The notation 4,~s-B,, in this lemma and in its proof, shall in- variably mean that 4,:B,—s, — even if s=0. 482 Chapter XIII. Divergent series. it remains valid when k is replaced by a larger, or p by an arbitrary, integer => 0 (or both simultaneously). Proof. It is obviously sufficient to show that a) If the relation (*) holds for a specific £ > 0 and pl, it remains true when p is replaced by p — 1, i. e. sok m+k—v\/p—1+v» (ntk+p £4 Ed k ) (7217) ses(” al is an invariable consequence of (¥), and conversely (**) implies (¥). b) If the relation (*) holds for some value of 2 > 0 and p=0, it remains true when k 1 is substituted for k. The statement b) is the permanence theorem for the C-process: Chia (s,) — s involves C, , , (s,) — s. Hence, by theorem 1, the statement is true. To prove a), transform the left hand side of (*) by writing LE a (Pl and splitting up the expression in question into two parts. Spar) 2 im pied LE fri) s, r=0 y=0 pA E+1 E/ntktl1—n\(p—1+4v CT The first of the two sums on the right hand side, which coincides with that on the left haad side of (**), we write Wl »=0 and the second then becomes CL + (En tert (12 The relation (*), if we divide by (Eiion, thus assumes Rt+p+1 the form E+p\ Y4409Y., x . yo [hthkty pirun, ner Gao! a nt + WL tp n p im : Bt+p+1 : while the relation (**), if we divide by Et) simply becomes +p => 8% That the first of these relations is a consequence of the second now follows immediately from 44, 2; that, conversely, the first involves the § 60. The C- and H-processes. 483 second follows, equally directly, from 65. In either case we put oi fe ” k+p lemma is established. % and in the second case also o = RE, Thus the TG eA be the Now let (z,) be any sequence and let z,’ ar corresponding arithmetic means; then AC BI rn drt) = er Whe Accordingly, the first of the three sums ~( CI i. e. the g+1 sequence (z,) is limitable C with the value ¢, if, and only if, the (r+a-+1 : Nal occur if, and only if, the relation Jr Teel) r=0 q q holds. We therefore have also the following q+1 third of the sums also ~(- )- By the lemma this will Corollary. If (z,) is limitable Cyyq with the value {, the sequence of the arithmetic means z,| = ee is limitable C, with the value {, and conversely. From this we obtain in a few lines the Proof of the equivalence theorem. In fact, the corollary just given shows that each of the % relations eh = =i, {s) 25s Gli) — C, pe = hy —>s is a consequence of any of the others; in particular, the first is a con- sequence of the last, and conversely, q. e. d. 4° 40 Other proofs of the equivalence theorem were given by Faber, G.: Uber die Cesaroschen und Holderschen Grenzwerte. Sitzungsber. d. Bayr. Akad., Math.-Phys. Klasse, year 1913, pp. 519—531. Schur, I.: Uber die Aquivalenz der Cesaroschen und Hoélderschen Mittel- werte. Math. Ann, Vol. 74, pp. 447—458. 1913. Hausdorff, F.: See the first of his papers wentiofied « on p. 478. The above proof is a somewhat simplified form of the one given by the author in the paper mentioned on p. 487. 484 Chapter XIII, Divergent series. . / After thus establishing the equivalence of the C-process and the H-process, we need only consider one of them. As the C-process is easier to work with analytically, on account of the explicit formulae 265, 3 for the SA ’s, it is usual to give the preference to it. We next inquire how far its range of action extends, i.e. what are the necessary conditions to be satisfied by a sequence in order that it may be limitable C,. Using the notation, which was intro- duced by Landau and is now generally adopted, z, = O (n%), « real, n to indicate that the sequence (3) is bounded, and x, = o(n®) to in- n dicate that (2) is a null sequence*', we have the following theorem, n which may be interpreted by saying that sequences whose terms in- crease too rapidly are excluded from C,-limitation altogether: 371. Theorem 3. If 2 a,, with partial sums s,, is summable C,, then n? a,=o(n") and s = on"). Proof. With the notation of 263, 3, the sequence of numbers SO SE=Dy...q SE C +R nk I at : : -k— +k is convergent. Since tg : 3 ~ ; ): the sequence 3 k—1 , (k—-1) Sy H..el Le, [= - a \ & is convergent, with the same limit. The difference of the two quo: : : oe k tients, viz. Sy We: )s therefore forms a null sequence. As 1 ~ n* this implies that Sy Y=o(n*). It follows that SE-3 So=0 — SEV =p (n*) +0 (n") == 0 (n¥), 13 41 The first statement thus implies that the quantities |a,| are of af most the same order as const..-#n%, the second that they are of smaller order than n*, in the way in which they increase to + 00. 42 The reader will be able to work out quite easily for himself the very simple rules for calculations with the order symbols O and o which are used here and in the sequel. § 60. The C- and H -processes. 485 and similarly SEV oll), oes Si=0ldl), 2, =0l07); a =o) For k= 0, this theorem reduces to the theorem 82,1, so that we are here dealing with the generalization of the latter. However, the intermediary result SJ" — o(#") just obtained in the proof may be interpreted as an even more significant generalization of the theorem in question. In fact, it means that (arenes lim £1) We accordingly have the following elegant analogue of 82,1: Theorem 4. In a series 2 a,, summable Cy, we necessarily have 272. C,-lima,=0. Moreover, even Kronecker's theorem 82,3 has its exact analogue, though we shall confine ourselves to the case p, = n: Theorem b. In a series 2a,, summable C,, we necessarily have 273. C,-lim pins) =0. In fact, it follows from the corollary to 70 that C,(s,)—s involves C,_, (pratt Th *)—s, and therefore by the permanence theorem C, EE . *) —s. Subtracting this from C,(s,) —s, we at once obtain the statement C, fim (5, — Sato t on) C tim (42st Sr =0 By means of these simple theorems, the range of action of the C,-process is staked off om the outside, as we might say, for the theorems inform us how far at most the range may extend into the domain of divergent series. Where this range properly begins is a much more delicate question. By this we mean the following: Every series coavergent in the usual sense to the value s is also summable C, (for every k > 0) to the same value s. Where is the boundary line, in the aggregate of all series which are summable C,, between convergent and divergent series? Here we have the following 486 Chapter XIII. Divergent series. Theorem 6. If a series 2 a, is summable C, and if its terms a, satisfy the condition : 1 a, =) (+) 3 then 2 a, is convergent. (O-C,— K-theorem)*3. Proof. By hypothesis Gr welll; : Noterot (2D i k By the lemma 270, we therefore also have, for every integer p > 0, A Ese eer ET ® ’ + E+ 3 k+p If we replace all the s’s by ss in the numerator, this expression simply reduces to s,. Hence we have == rte 2 YE Bait Now, by hypothesis, (na,) is bounded, say |= a l< RK. Therefore we have for y=0, 1, ..., #— 1, +1 js, —s,[=]0,., 1 -+a,| KK] =H -1 and therefore Is, — 0 | Shen EE CITE - 20) To nrg m (11521) 111) st de smd = 1521) n= Ete BCE) - CEI 43 Hardy, G. H.: Theorems relating to the convergence and summability of slowly oscillating series. Proc. Lond. Math. Soc. (2) Vol. 8, pp. 301—320. 1909. — The theorem deduces convergence (K) from C-summability. We -accordingly call it a C— K theorem for short, and more precisely an O-C—>K theorem, since an O (that is, the boundedness of a certain sequence) is employed in the determining hypothesis. A theorem of this kind was first proved by A. Tauber, — in his case, for the A-process (v. 286); for this reason, Hardy gives the name of “Tauberian theorems” to all theorems in which ordinary convergence is deduced from some type of summability. - \ § 60. The C- and H-processes. 487 Keeping p fixed and letting #n — -- 00, it follows, since 6 —s, that Tole ~ sig BEL == As this must hold for every p, however large, it follows further that mis —s|=0, |e s,s, gq ed J 0 0 1 3 Application. The series 2 == 2 Trai «=z 0, is not convergent, n= n= as it is easy to verify, by an argument modelled on the proof on p. 442, foot- nete 56, that for n=1,2,..., wl Ald at hn adrtel fa nba 1)” n? with (9,) bounded. Further, for this series (na,) is bounded, hence the series cannot be summable Cj to any order. Closely connected with the preceding, we have the following theorem, where for simplicity we shall confine ourselves to summation of the first order. Theorem 7. A necessary and sufficient condition for a series 2a, , 275. with partial sums s,, to be summable C, to the sum s, is that the series (4) Ay Ds v=1" should be convergent and that if o, denotes its partial sums and o its sum, the relation s—s,=mn(c—o,)+o(1) holds, 1. e. B) Sok (nb 1) [mr pop] 7" Proof. lf Za, is summable C,, we have by 183, since a, = S,. or S, = 2 i Es Bet nin veil lL atl VT <= vo+1) " atp+1’ 4 Knopp, K.: Uber die Oszillationen einfach unbestimmter Reihen, Sitzungsber. Berl. Math. Ges., Vol. XVI, pp. 45—50. 1917. : Hardy, G. H.: A theorem concerning summable series. Proc. Cambridge Phil. Soc. Vol. 20, pp. 304—307. 1921. Knopp, K.: Zur Theorie der C- und H-Summierbarkeit. Math. Zeitschr. Vol. 19,-pp. 97—113. 1923. 488 Chapter XIII. Divergent series. and, since s, = S,’ — S,/_,, on again applying Abel's partial summation this becomes ; Sn Si 7+t1 (vy 1) (n+ 2) 2 n+p Sa in n+p Ste + Tren y Sn TEI her As n— 400, all five terms of the right hand side tend to O, whatever the value of p, for by the assumed C,-summability and theorem 3, s, =o (un) and S,' = O(n). Therefore I= ts convergent. At the same time, keeping » fixed and letting p — -- co, we obtain S$, ’ Sab) 3 = Pasb2e 1) = SEIDGTY’ r=n+1 As this tends to — s--2s=s as n—00, (B) is true. If, conversely, the condition (B) is fulfilled, — which implies the convergence of z=, — then by the expression just obtained for the left hand side of (B), we have Ss, J sentra ty 2 rly hol) or / S,/ i — Su 4 2 fo 1)(n + s(n 4 2) + o(n) =s-n-+o(n). Writing for short 0 S. ’ 1) (nL 2 ra? ew Cede ts 2 erie our assumption (B) therefore takes the form (B) : 20,— SS, =s-n-+o(n). Now 0, 0 T= = {7 he 1) (n - 2) > a 5; Ss, 3 — n(n -1- 9; nnt+1)(nL+2) + 2 wie 2 On Se CH 2 On — Sul iE eT3 may By the relation (B’) just obtained, we therefore have 8, = yy = So 0(1s and by 43,2 it follows at once that : 0, =s-n + o(n). § 60. ‘the C- and H-processes. 489 Substituting this in B’, we finally obtain SS, =s-n-o0(n), 6 = =s+o(1), ie c'—s. Za, is therefore summable C, to the value s *>. We shall content ourselves with these general theorems on C- summability 6 and we shall now proceed to a few applications. Among the introductory remarks (pp. 461-462), it was pointed out that the problem of multiplication of infinite series, which remained very difficult and obscure as long as the old concept of convergence was scrupulously adhered to, may be completely solved in an extremely simple manner when the concept of summability is admitted. For the second proof of Abel's theorem (p. 322) provides the Theorem 8. Cauchy’s product 2c, = 2 (ayb,+a,b,_, +--+ +a,b,) 276. of two convergent series 2a, — A and 2b, = B is always summable C, to the value C= A-B. Over and above this, we now have the following more general Theorem 9. If Xa, is summable C, to the value A and 2b, is B77. summable Cy to the value B, then their Cauchy product Ze, =Zla, 0,1 ab, +t -+1a,b) is certainly summable C,, to the value C=A4-B, where y = a+ 4-1. Proof. Let us denote by 4%, BY, CY the quantities which in the case of our three series correspond to the S's of the general C-process as described in 265, 3. For |x| < 117, since n Wr, n 2a,5% 30, 8" =e 3", we have 1 Az)" tt Hence, by 265, 3, Cl =APB2 + ABE +. + 47 BY. It follows at once from this that (r+ Ca’ ? )—4 2 as asserted; for, quite analogously to 48, 6, we have the following general theorem, where we make use of the notation introduced in 270: . 45 The theorem may similarly be established for Cj-summability; cf. the third of the papers mentioned in the preceding footnote. 46 A very complete account of the theory is given by Andersen, A. F.: Studier over Cesaro’s Summabilitetsmetode, Kopenhagen 1921. 47 Since a,=0(n%), b,=0 n#, the power series employed are absolutely convergent for |x| <1. Ba atti Th mommies 2p of, Qd-a)tt d—p)rtt n 27S. 490 Chapter XIII. Divergent series. Auxiliary theorem. From x o£. i 4 and y,29- & ; 2 it follows in every case that Var Ly Ynt + HW YD En. ile with y=a-+ +1. Proof. Putz, = (£44) 483 3 and y, = (n+ 9,) 25 , where, by hypothesis, §, —0 and 4, —0. As Gel ee OL 000 it is sufficient to show that 0,— 0 and J — 0 always imply that the expression n+ o\ /f n—1+4+a\/1+6 i a\ n+p 02, ("7% (+08 TW) +00 07) divided by Cr , tends to 0. Now given an arbitrary ¢ > 0, choose 7, so that, for every » > 5 d #,> |8,| and |8. | are n,, £390 ++ Q(T oT) From this our statement follows at once. Examples and Remarks, 1. If the series X'(— 1)” is multiplied by itself (# — 1) times in succession, we obtain the series ¢=1 21 1 uly 3 G=1,2..) The original series (2=1) i summable C, by 262, its square is (cer- tainly) summable C;, its cube summable C,, etc. However, by 2682, we know that the kt of these series is (exactly) ‘summable Ck. 2. These examples show that the order of summability of the product- series given by theorem 9 is not necessarily the exact order, and that in special cases it may actually be too high. This is not surprising, inasmuch as we already know that the product of two convergent series (k =0) may still be convergent. The determination of the exact order of summability of the pro- duct series requires a special investigation in each case. In conclusion, we will investigate one more theorem which may be materially extended by introducing summability in place of con- vergence, — namely Abel's limit theorem 100 and its generaliz- ation 233: Theorem 10. If the power series f(x) = Za, 2" is of radius 1 and is summable C, to the value s at the point +1 of the circumference of the unit circle, then fBD=(¢)>s; § 60. The C- and H-processes. 491 for every mode of approach of z to 1, in which z remains within an angle of vertex —-1, bounded by two fixed chords of the umit circle (v. Fig. 10, p. 406). Proof. As in the proof of 2833, we choose any particular se: guence of pointe (z..2,,..., %iy--.) within the unit ciicle and the angle, and tending to --1 as limit. We have to show that f(z;)—s. Apply Toeplitz theorem 221 to the sequence mam STH) 45 which by hypothesis converges to s, using for the matrix (a;,) a (a) (1 — z)kt1.27". We deduce at once that the transformed sequence also tends to s: 18 o = =n 0 n = (1 = Jort, Ss 2s ’ Since ISP "= rf), this is exactly what our statement implied. For this proof to be correct, we have, however, still to verify that the chosen matrix (0) satisfies the conditions (a), (b) and {c) of the theorems 221. Since z;,— 1, this is obvious for (a); and, since 0 DL In + k 1 A, = 2 Pn = (1 —_ we ( k Yor == (1 —5) tp —z)k+1 =1, (c) is also fulfilled. The condition (b) requires the existence of a constant K’ such that Zl, ={ot ei 2 | for every 1. By the considerations on p. 406, this is obviously the case with K’ = K*+1 if K has the meaning there laid down. For k= 0, this is exactly the proof of Abel's theorem as carried out on pp. 406—407, in the generalized form of Stolz. For k=1, we obtain an extension of this theorem, first indicated by G. Frobenius*?, and for k= 2, 3,... we obtain further degrees of generalization, due in substance to 0. Hdlder®® — taking H,- instead of C,-summability and approaching along the radius instead of within the angle only — and first expressed in the form proved above (though with entirely different proofs) by E. Lasker and A. Pringsheim®® 46 k is now the fixed order of the assumed summability. 4 Journ. f. d. reine u. angew.- Math. Vol. 89, p. 262. 1880. 0 Cf. the paper cited on p. 465, footnote 12. 51 Phil. Trans. Roy. Soc., Series (A), Vol. 196, p. 4381, London 1901. 52 Acta mathematica Vol. 28, p. 1. 1904. Prd! 492 Chapter XIII. Divergent series. By this theorem 10, we have, in particular, lim (Za 2") =, for real x's increasing to + 1, and accordingly we can express the essential content of the theorem in the following short form, which is more in keeping with the context: : Theorem 11. The C,-summability of a series Za, lo a value s always involves its A -summability to the same value. 2 With the exception of the C,-summation of Fourier series, which will be considered more fully in the following section, further applications of the C,-process of summation mostly penetrate too deeply into the theory of functions to permit us to discuss them in any detail. We should, however, like to give some account, without detailed proofs, of an application which has led to specially elegant results. This is the application of C,-summation to the theory of Dirichlet series. The Dirichlet series Z (-1)~1 45 f= FED n=1 n? is convergent for every z for which %R(z) > 0, divergent for every other z. At the point 0, however, where it reduces to the series 2(— 17% it is summable C, to the sum I at the point — 1, n=1 0 where it reduces to Y(— 1)?~1z, it is (cf. p. 465) summable C, n=1 to the sum 3 and the indications given in 268, 3, show that for $. 2 = — (k — 1) the series is summable C, to the sum Xela, for every integral value of k > 2. This property of being summable C,, for a suitable %, outside its region of convergence R (2) > 0, is not restricted to the points mentioned; it can be shown by relatively simple means that our series is summable C, for every z with $i (z) > — k. Moreover the order of summability is exactly k throughout the strip =< RL ~{5—1). Thus in addition to the boundary of convergence, we have boundaries of summability of successive orders, the domain in which the series is certainly summable to order % being, in fact, the half-plane ND>—4,; (h==0,12,..) 2 f{7)= (a3 -£ (2), where 0-3 is Riemann’s {-function n= (Cf. 256,4,9,10 and 11.) § 61. Application of C,-summation to the theory of Fourier series. 498 Whereas formerly it was only with each point of the right hand half-plane % (2) > 0 that we could associate a “sum” of the series wr: n—1 So entire plane, thus defining a function of z in the whole plane. In a way quite analogous to that used for points within the domain of convergence of Dirichlet’'s series, further investigations now show that these functional values also represent an analytic function in the domain of summability — i. e. in the whole plane. Qur series therefore defines an integral funciton®. Quite analogous properties of summability belong in general to every Dirichlet series , We now associate such a sum with every point of the J 9 5s n=1 n? Besides the boundary of convergence R(z) =1, — or A,, as we shall now prefer to write, since convergence coincides with C,- sum- mability, — we have the boundaries $i (z) = 1, for C,- summability, k=1,2,.... They are defined by the condition that the series is certainly summable to the kth order for R(z) > 1,, but no longer so for (2) <1. We of course have 4, >1, >1,>---, and the numbers 4, therefore tend either to — co or to a definite finite limit. Denoting this in either case by A, the given Dirichlet series is summable C, for every z with R(z) > A, where k is suitably chosen, and its sum defines an analytic funciion which is regular in this domain. If A is finite, the straight line % (2) =A is called the boundary of summa- bility of the series. For the investigation of the more general Dirichlet series ae; it has been found more convenient to use Riess R,,- summation, Cf. the tract by Hardy and Riesz mentioned in 266, 2. § 61. Application of Ci-summation to the theory of Fourier series. The processes described above possess the obvious advantage of all summation processes, namely, that many infinite series which pre- viously had to be rejected as meaningless are henceforth given a useful 5 From this it follows fairly simply that for Riemann’s {-function the i 1 4 ; : difference (== is an integral function, — an important result. 5 Bohr, H.: Uber die Summabilitit Dirichletscher Reihen, Gott. Nachr. 1909, p. 247, and: Bidrag til de Dirichletske Rikkers Theori, Dissert., Kopen- hagen 1910. 17 280. 494 Chapter XIII. Divergent series. ‘meaning, with the result that the field of application of the theory of infinite series is considerably enlarged. Apart from this, the extremely satisfactory nature of these processes from a theoretical point of view lies in the fact that many obscure and confusing situations suddenly become very simple when these processes are introduced. The first example of this was afforded by the problem of the multiplication of infinite series (see p. 461, also p. 489). But the application of C,- summation which is, perhaps, the most elegant in this respect, as well as the most important in practice, is the application to the theory of Fourier series, due to L. Fejér 3%. As we have seen (pp. 369—370), the question of the necessary and sufficient conditions under which the Fourier series of an integrable function converges and represents the given function is one which presents very great difficulties. In parti- cular, it is not known e. g. what type of necessary and sufficient con- ditions a function continuous at a point z, must satisfy at that point in order that its Fourier series may converge there and represent the functional value in question. In § 49, C, we became acquainted with various criteria for this; but all of these were sufficient conditions only. It was for a long time supposed that every function f (x) which is con- tinuous at x, possesses a Fourier series which converges at that point and has the sum f(x,) there. An example given by du Bois-Reymond (see 16,1) was the first to discredit this supposition. The Fourier series of a function which is continuous at x, may actually diverge at that point. The question becomes still more difficult, if we require only — as the minimum of hypotheses regarding f(x) — that the (integrable) func- tion f(x) should always possess unique right hand and left hand limits, f(x,+ 0) and f(x, — 0). What are the necessary and sufficient conditions which must be fulfilled by f(x) in order that its Fourier series may converge at x, and have the sum 2 [f+ 0) + f(x, — 0)]? As was pointed out, this question is not yet solved by any means. Nevertheless, this obscure and confusing situation is cleared up very satisfactorily when the consideration of the summability of Fourier series — C,-summability is quite sufficient — is substituted for that of hel convergence. In fact we have the following elegant Theorem of Fejér. If a function f(x), which is integrable in 0 +0 t—>+0 should exist separately; it is sufficient that ww A fim [f (5+ 0) + (@— 1] = 5 () t—>+0 should exist (cf. p. 370). 58) Here the arithmetic mean of the numbers s, is denoted by 0, =o, (2) instead of by s,’ =s,/ (x), to avoid confusion with the notation for different- ation. 496 Chapter XIII. Divergent series. to this and to the fact that the whole is multiplied by 2 the success of the subsequent part of the proof is due. If the two SE f(x, 0) and f(z, — 0) exist, or even (v. footnote 57) if only 1 im <5 [f(z,+ 28) + f(@— 20) = 5 (¥) = s t>+0 exists, Fejér's theorem simply states that ¢, — s. We observe that sin nt sin ¢ 2 ) di=nZ © my 151 since the integrand is 2 sin (2v—1)¢ sin ¢ r=1 and each term of this, when integrated from 0 to = contributes the value = — since sin (2 — 1) ¢ =1-}2cos2¢}-2cos4t4---42cos2(»—1)¢ %% sin ¢ Hence we may write 2 (= “ S - at na sin 4 0 and therefore 281. wl a Cp-1 sm [fro remty RL sin ¢ By hypothesis, the expression in square brackets tends to 0 when i—-10. In order to prove that ¢,_, or o,—s, it is therefore sufficient to show that -3 If ¢(t) is integrable in O... = and t>+0 then z 2 2 sin nt 2 [r0- (BR) ar—o0 as mn increases. d 5 The value of the integral may also be inferred directly from Fejér's integral itself, for f(z)=1, for which a,=2 and the remaining Fourier con- stants = 0. § 61. Application of C,-summation to the theory of Fourier series. 497 Now this follows from a very Site train of inequalities. As @(t)— 0, we can determine 5 cr , for a given £>0, so that lo @)| << for every ¢ such that 0 << ¢< 8. Then 8 8 Zle0- (aus s Ef (East since the last integral has a positive integrand, and therefore remains less than the integral of the same function over the whole range 0 to 3 On the other hand, a constant M exists such that |g (¢)] remains << M throughout 0 < ¢ < e . Consequently J 2 sin i 1 2 Jo0-( (Z) at <2 7 "sind * 8 On the right hand side, everything but # is fixed, and we can there- fore choose 7, so large that this expression becomes << = for every nm >n,- We then have Yo, 5 = s] << for these n’s; hence ¢,—s. Thus Fejér's theorem is completely established ©. Corollary 1. If f(x) is continuous in the interval 0 <2 < 27, 282. and if further f(0) =f (2x), then the Fourier series of f(x) is sum: mable C, to the sum f(x), for every zx. For he hypotheses of Fejér's theorem are now certainly fulfilled for every z, and 1 g [f (x40) + f(x — 0] ={ (x) everywhere. We assume, as usual, that yd function f(x) is de- fined in the intervals 2 kz 0, we can determine one number N such that for every nm > N, irrespective of the position of x, we have |e, (®) — f(@)]| < 2 % Note in passing that the curves of approximation y =g, (x) do mot exhibit Gibbs’ phenomenon (v. 216, 4). (Fejér, L.: Math. Annalen, Vol. 64, p- 273. 1907) ® The corresponding statement holds, moreover, in the case of the general theorem of Fejér for every closed interval in which f(z) is continuous. 498 Chapter XIII. Divergent series. Proof. We have only to show that the inequalities in the proof of the theorem can be arranged so as to hold for every x. Now #() = pt) =5[f@@+20 — F@] + 5 [fo —20) — F@)]; since f(x) is periodic and is continuous everywhere, it is uniformly continuous for all 2's (cf. § 19, theorem 5), and, given ¢, we can choose one é > 0 such that Ifex2)— fl) <5 for every |¢| < J, and every x. This implies that for all these fs &€ le) =e) <4 irrespective of x; hence, as before, 8 2 sin n#\2 8 0 Further, since f(x) is periodic and is continuous everywhere, it is bounded, say |f(x)| < K for every x. It follows at once that for all gs and all u's, le@)| =| 2)|<2K and hence, as before, 9 sin n#\2,,1 1 2K = fn (22 2S am 0 Now we can actually determine one number N such that the last ex- wo] 4 . . & ’ ! pression remains << for every n > N. For these n’s we therefore have Joss — S$ < ¢&, so that, as asserted, we can associate with every given ¢ one number N such that \ l0,(2) — flR)| N, irrespective of the position of z. § 62. The A-process. The last theorem of § 60 has already shown that the range of action of the A-process embraces that of all the C,-processes. In this respect it is superior to the C- and H-processes. Also, it is not difficult to give examples of series which are summable A but not summable C, to any order k, however large. We need only con- sider 2a, «", the expansion in power series of 1 (i) = et § 62. The A-process. 499 at the point x=—1. Since obviously lim f(x) exists for x ——1-4-0 and = Ve, the series X (—1)"aq, is summable 4 to the value Ve. If, however, it were summable .C,, for some specific k, by 271 we should require to have a, = o(n"). Now a particular coeffi- cient a, is obtained by adding together the coefficients of 2" in the expansions of the individual terms of the series, which is uniformly oavergent for lz] p< 1: Sot oF Ta +o aot (v. 249). As all the coefficients in these expansions are positive, a, is certainly greater than the coefficient of 2" in the expansion of a single term. Picking out the (k + 2) term, we see that 1 nt+k41 nkt1 *k+ 2)! 1 E41 )> C+ G+” For a fixed %, a, | un” therefore cannot tend to 0; on the contrary, it tends to -} co. Although the A-process is thus more powerful than all the C, processes taken together, it is, nevertheless, restricted by the very simple stipulation that in order that it may be applicable to a series Sa, , the series Za a" and X's, 2" must converge for |x| <1: a4, == Theorem 1. If the series 2a,, with partial sums s,, is sum-283. mable A, we necessarily have imV]a, [<1 and Im VTs,] or, what comes to exactly the same thing, a,=0(1-+¢") and = 014, for every & > 0, however small. In this we have a companion to theorem 3 of § 60; but theorems 4 and 5 of that section also have literal analogues in this connection: in Theorem 2. In a series 2a,, which is summable A, we neces- 84. sarily have . oo 3 : a, +2a,+---+na, A-lma,=0 and indeed A-lim (rtd vo i ) =i) Proof. The first of these two relations indicates that (1—) 2a, 2" must tend to 0 as x —1— 0. This is almost obvious, since by hypo- thesis Xa, a" —s. The truth of the second statement follows, on the same lines as the proof of 273, from the two relations Sok Sie t sy *) Q—=2)-2s,a"—s and (Ler nf) BREA LE 20 es by subtraction; the first of these is nothing more than an explicit form of the hypothesis that Xa, is summable A, while the second is 285. 500 Chapter XIII, Divergent series. quite easily deduced from it. In fact, from (1 —2)Z's a" os swe first infer that Q—2?23(sy+s, + Fs,)2"—s, by 102. That the second of the relations (¥) follows from this, is a special case of the following simple theorem: Auxiliary theorem. If, for x—1—0, a function f(x), which is integrable in 0 0, we can choose an x, in 0 <<, <1, such that |p (z)| remains < = foro, <<. As A io + [tan i 1 we therefore have, for these values of x, — if I, denotes the (fixed) value of the first of the two integrals on the right hand side, — 0 — FD) =| SO —a)-[s| +09): 4] +5 Now choose z,, with &, < x, < 1, so that 1 — a) (|s|+ ||) <2 for Z,< <1. Then, for these values of =z, (1-2) Fx) —s] 0, we can choose n,> 0 so that for every n> n, a) na, <<, b) gto alpen] oo 3 3 5 Fli—1)—s| <5 (F(2) = Za,a"). (Here a) and c) can be satisfied by hypothesis, and b) by referring to 43, 2.) For these u's and for every positive x <1, we then have n £ 0 s,—s=f@)—s+ Ya Q—2")— 3 az’ y= r=n+1 If we now observe that in the first of the sums fr) =item driing,] Lol _ and in the second [gq |= << oy it follows that Js, —s|<1f@) —s| +12). Siva, | wae for every positive @ <1. Choosing, in particular, s=1- 1, we obtain, by a), b) and c), js, —s]l <2 r+ 3 + 5 =e for every n> n,. Hence s,—s, qed On account of the great similarity between this theorem and theorem 6 of § 60, it appears likely that an 0-4 — K-theorem also holds, i. e. one which deduces the convergence of 24, from its 4-summability, by assuming, in connection with the a's, merely the fact that they are 0 (5): This theorem is actually true. It goes very much deeper, however, and was proved for the first time in 1910, by J. E. Littlewood ®*: Theorem 4. A series 2 a,, which is summable A, and whose 287. terms satisfy the relation 1 a, =0 (5) ) — i.e. for which (na, ts bounded, — is convergent in the ordinary sense. (0-A— K-theorem.) 62. The converse of Abel's theorem on power series: Proc. Lond. Math, Soc. (2) Vol. 9, pp. 434—448. 1911. = 17% 502 Chapter XIII. Divergent series. Proof®s, ~ L Under the present hypothesis to the effect that (na,) is bounded. the relations a), b), c) of the preceding proof continue to hold — and indeed for every m —, provided ¢ is interpreted as a suitable, i. e. possibly large, positive constant. That proof accordingly shows that with the present hypotheses |s, —s| 1: (t— of @| <6, If im f(x) exists for x—1 — 0, then, for every integer q > 1, 1—2)02f9@x)—0 as z—1—0. This is true for g=1 by the preceding lemma. We proceed inductively: we assume that the statement has been established for g=1,2,....k(k 21) and deduce its validity for g=% +1. Now, in 0< o< 1, writing 1 — a) r®(@) =g@), we have g =~ 2{i ~ a) {OF (1 ~ B)ifE ED), (2) = h(E — (1 — 2)=2 (2) — 2h(1 — ab=1 f+ 2) oo 1 -— z)% f+) {@). Accordingly, g(x) satisfies the conditions of the previous lemma, since lim g(x) exists for x—1— 0 and 1-2)" (@)| Sk(k—1)G, + 2%G,y; + Gy =. Hence, by that lemma, {1 — )g’ (x)—O0, ie. Lz (1 = 2)" fo (x) 4 (1 a wy f&+1) (x) — 0. Since here the first term on the left tends to 0 by the assumption of our proof by induction, it follows that the second does also. Thus our statement is established. IV. In virtue of this lemma, we may now deduce from the hypo- theses of theorem 4 a conclusion which is decisive for our present purpose, and which corresponds exactly to the passage from c} — s 10 02 -» 3 (see p. 486) in the proof of the O-C-— K-theorem. We show that: The A-summability of 2 a, to the value s, i.e. the relation oO lim (1 — x) Pid =s, z->1-0 n=0 together with the boundedness of the sequence (na), Ly : na, | 0, li ATELY n+p Yo i ®) oll ee di This we again prove by induction. For p= 0 the statement is identical with the assumption (*). Now let k denote a definite integer = 1 and assume the statement proved for every p < k — 1. Its validity 504 Chapter XIII. Divergent series. for p=~F% may be inferred as follows: f(x) = Ya 2" implies, for every 7 > 1, that n SFO (@) = nig) : Ne n=0 hence 1 K = /5-l7-1 K ¥ wool IT N=, RB if Nad 2 r—1 ? Al —n) Accordingly, (1 — )7 f(z) is bounded in 0 1. Further, since by the hypothesis (*) lim f(x) for x—1—0 exists, we have by III, for every integer » > 1, (1 — a) f@ (x)— 0, and, in particular, (1 —=2)*f® (z)— 0. Now ak 2 a= tS 21 — x) 2s, z"] h—1 n =~ gp San? =¢ P-1 V5, fon : Hence the above conclusion implies that x n+ k nt+k—1 =p 2 1 Yount == {1 > 2) PA 1 Vintwy 0. As here the second term tends to s by the assumption of the proof by induction, the first must also —s. Thus (}) is proved in general Proof of theorem 4. Let ¢ > 0 be given arbitrarily. We shall show that a number mm, exists such that |s, —s| <&(K- 5) for every m > m,, where K denotes a number greater than all the numbers |na, |, the existence of which is ensured by the hypothesis that (na) is bounded. An m, of the required type may be obtained as follows: & a) Write os e™° = and Te =, so that & and 7 are posi- tive but <1 (by 114). Hence we can determine a positive integer p for which 2Mpov ho the following conditions are fulfilled: 4 (c,) Tar (cy) > == (CG) p 2s Spill 7 s a +1 = np a n 4 6! x)? = p Ys. Sutp) © 2 hence, by b), (5) ls, —sl (1?) a =e, n=0 for every z, m,.- In the next instance, we have for 2; and 2 2 (<2 2 P10 | 2s <2M- > C1 p n=my—p In these sums, the nt term results from the (n — 1) by multiplying the latter by the factor 9 et n-1. EP Zz" xT = —"z. (7 ’ , Hence the terms increase or diminish as # increases, according as AV 2221. In order to have the terms all increasing in the first sum, and all decreasing in the second, we choose m mp’ a value <1, but > x, by (c,), and we now keep this value of x fixed. As the last term is now the largest in the first sum, we have ee Ye hs) 2, = a fre Let us substitute Om, — 1)? for (72 H and increase the expression xT — further by taking as base instead of (m, — 1) and replacing 1, m 11+ by i in the exponent. Our expression is thus m ppt+1 1 m \1+s SE We continue to increase the right ond side by substituting e? for 2 (14)? for (1+ ¢)?+! and ¢ “ats for om 1 onl; hus gl’ mp m-+p we now have : pis gh kL © 2M-p e ve Lie (tomin) (1+)? By a) and (c,) we therefore infer that " The latter because 1 — a << e © for «40 (see 114). § 62. The 4-process. 507 We may evaluate 2, the terms of which diminish monotonely, in quite a similar way: Since the factor ze 2, which when multiplied by the (n — 1)th term gives the nth term 0 above), itself decreases monotonely as # increases, the sum 2 is certainly less than the geo metric series wep rf. Pat? Y 2M("p) 2h the terms of which are obtained by continued multiplication by the first (i. e. largest) of the factors in question, starting with the first (i. e. largest) term of 2,. This geometric series is also convergent in : # si Hi, : reality, for the fact that Pry diminishes monotonely as # increases means that (myg+1—p)+p my—+1—p m-+p m 1 << — = x provided only that m, +1—p > m. This is actually the case, by (c,) and (c,). It follows that 2 \2hl m m \My—p 1 i 41. > (2). . 1 x)? 15l<(; =r) 2 M-("y) = Hh tl m mgt 1—p mp. We increase the last fraction on the right hand side further if we substitute m (1 4 ¢) for m, + 1 in it; for Tre increases when y dimin- ishes. Doing this and also substituting (m (14g)? for fi and m(1-+e)—1—p for the exponent m, — p, we obtain, after a few slight transformations, the expression: ; pP+1 mp m m (1+¢) 1 rt eS — ry mle) —p This may be further increased, exactly as before, by substitiing ep id PP Tmtp m : i for 1 and e for Fa Observing that the last fraction of the right hand side pdt-p m (1+ ¢) 1 142 1 pme—p? pme—p) Pp 1.2 ? this gives p2(1 +e) 1 1 2.4 + &)? (e pl sC nt ) £3 — By a) and (c,) this again gives Q—zprt. |Z |< 2. < 2 M-eo 2 28S. i — 508 Chapter XIII. Divergent series. Combining the results and substituting in (¥,*), we obtain = ls, — s| < (K+ 5) for every m > my. This completes the proof that s_ converges to s. Examples and Applications. 1. Every series which is summable C is also summable 4, to the same value. This often enables us to determine the values of series which are sum- mable C. Thus in 268,3 we saw that the series X(—1)* (n+ 1)* are sum- mable Cj. q,; by means of the 4-summation process we can now obtain the values of these series. For #>0, e~! <1, so that the series lr aa is convergent and et or elt 152 od 2 te tify] e¥ 1 eh] 1 1 ¢ 1 9 Pela 1s My For a sufficiently small > 0, these last fractions may be expanded in power series by 1095, 5; the first terms of the two expansions cancel each other and we obtain 0 wt ele pp (mre mls seh (e411 Differentiating % times in succession with respect to #, we further obtain ei-9% 200 A ( Dot He Br... x orl 1 —k = rf Be =D (r= oR De" Now, letting # diminish and 550, we at once obtain on the right hand side ghetto Ei err Brat a Putting e~% =z on the left hand side, we see that we are dealing with a power series of radius 1; when ¢ decreases to 0, x increases to 4-1. The value just obtained is therefore by definition the A-sum of the series 1-254 8% dee (=) (n+ 1)* foe for integral 2>0. And as this series was seen to be summable C;.4 in 268, 3, we have thus obtained its Cj q-sum also, by 279. 9. If the function represented by a power series Zc, 2" of radius 7 is regular at a point z, of the circumference of the unit circle, lim f(zz) for positive increasing @ — 1 certainly exists and = f(z). At every such point the series Ja, = X¢,z2" is therefore summable 4 and its 4-sum is the func- tional value f(z). Z 7 For k>> 0, the sign (— 1)¥*! may simply be omitted, by the footnote on p. 237. § 63. The E-process. 509 8. Combining the preceding remark with theorem 4, we get the state- ment: If f(x)=2¢,2" converges for |z| <1 and (nc,) is bounded, then the series continues to converge at every point z,, on the circumference of the unit circle, at which f(z) is regular. 4. Cauchy's product Yc, = 3 (ayb,-+-.-+a,b,) of two series Xa, and 2'b,, which are summable 4 to the values 4 and B, is also summable 4, to the value C = 4 B, as an immediate consequence of the definition of A4-sum- mability. 5. With regard to the series >’ n=1 in 274 that these do not converge, and that they are not summable Cj to any order k. By Littlewood’s theorem 4, we may now add that they cannot be summable 4 either. 1 es =Z Ww 1 y iin = 0, e have a ready seen § 63. The E-process.” The E,-process was introduced on the strength of Euler's trans formation of series (144). Starting from any series Xa, (not having alternately + and — signs), we should have to write 1 n n n 73 (0) tt (a+ + (0) =a! and we should have to consider Xa as the E, -transformation of 2a,. We had agreed to depart from the usual notation so far as to write s,=0 and s =a, +a, +4: ta. for. 4>0%, — and similarly for the accented series. Then (v. 265, 5) w= [Dn att ()e] is the E, -transformation of the sequence (s,). Applying this again, we obtain for the E,-transformation, after an easy calculation, Za, with af =p [(G)8" a+ (1) 3" a toe (3) a] the partial sums of which are now By Tal ebay =3,." ~H[Emet (eat (Jal, 620 2 A detailed investigation of this process is to be found in two papers by the author, Uber das Eulersche Summierungsverfahven (1: Mathemat. Zeitschr. Vol. 15, pp. 226—253. 1922; II: ibid., Vol. 18, pp. 125—156. 1923). Complete proofs of all the theorems mentioned in this section are given there. 3 Jt may be verified without much difficulty that in the case of the E- process “a finite number of alterations” is allowed, as in the case of con- vergent series. (A proof, into which we shall not enter here, is given in the first of the two papers mentioned in the preceding footnote.) Conse- quently the shifting of indices has no effect on the result of the limitation process. 510 Chapter XIII. Divergent series. and for the E ,- transformation we in the same way obtain the series (p) ; By with terms 2 i an (0) (2” = 1" %o i (1) 2? T ore ay =~ TR = & a,) and partial sums (# > 0) aD ai lal el = IE Wan 0 inde 00a The examples given in 263, 5 have already illustrated the action of the E, -process; the last of them shows that the range of the E,-process is considerably wider than those of the C- and H-processes. By analogy with that example, we may form the E -transformation of the geometric series 22";-and we shall obtain Ht S00 wee] 3 Sani ; . = 1 ‘ . This series converges”, — to the sum oe if, and only if, |z 4 (2% —1)| < 2%, i e. if z lies within the circle of radius 2? round the point — (2? — 1). Evidently every point in the half-plane % (2) <1 can be made to lie inside such a circle, by taking the exponent p sufficiently large. We may accordingly say: The geometric series 22" is summable E, to a suitable order p for each point z interior to the half-plane (2) <1. The sum is in every case ,——, i e. it is the 1 analytical extension of the function defined by the series in the unit circle. The case of any power series is quite similar, but in order to carry out the proofs we require assistance from the more difficult parts of function theory. We shall therefore content ourselves with indicating the most tangible results: The power series Yc, 2” is assumed to have a finite positive radius of convergence, and the function which it represents in its circle of convergence is denoted by f(z). This function we suppose analytically extended along every ray am z= @ = const. until we reach the first singular point of f(z) on this ray, which we shall denote by {,. (If there is no singular point at all on the ray, it may be left entirely out of account.) For a particular integer p > 0 we now describe the circle = 292.9 ® which corresponds to the one occurring in the case of the geometric series, and which we shall denote by K,. The points common to all the circles K ~ a “ The formula of this transformation suggests that, for the order p, the restriction to integers => 0 might be removed. Here, however, we shall not enter into the question of these non-integral orders. % This series is then, moreover, absolutely convergent. " As regards the proof, see p. 509, footnote 72. EVI ROY § 63. The E-process. 511 will, in the simplest cases (i. e. when there are only a small number of singular points), make up a curvilinear polygon whose boundary consists of arcs of the different circles, and in every case they will form a definite set of points which we denote by ¢&,. We then have the Theorem. For each fixed p, 3c, 2" is summable E, at every intevioy point of 289. ®, and the E,- transformation of Ze¢,2" is indeed dally convergent at that point. The onericel values thus associated with every interior point of ©, form the analytical extension of the element 2c, z" into the interior of &,. Outside &,, the E,-transformation of Xc,z" is divergent. After noting these examples, we now return to the general question, within what bounds the range of action of the E-process lies; in this as in further investigations, we shall restrict ourselves to the first order, i. e. to the E -process. The cases of E-summation of higher orders are, however, quite analogous. Since E,-summability of a series 2a, means, by definition, the convergence of its E, -transformation Zeng) +--+ ()a]. the general term of the latter must necessarily tend to 0: La (Jo++ (a0 which we may now write, for short, E, lima, =0, In this form, we again have an exact analogue to 82,1 and 272 or 284. Kronecker's theorem 82, 3 also has its analogue here. For if (s,) is a sequence which is limitable E, with the value s, its E,- transformations E, (s)) =s,’—s. The arithmetic means tlt. ts n+ 1 of the latter therefore also tend to s; we may denote them for short by C,E,(s,), since they are obtained by applying in succession first the E,-transformation and then the C,-transformation. Now it is easy to show by direct calculation -— we prefer to leave this to the reader —, that we obtain exactly the same result if we apply the C, -transfor- mation first, and then the E,-transformation, i. e. if we form the sequence Z. C, (s )=E£, [hh We have C, E, (s,)=E,C (s.); the two transformations are completely identical”. Thus we also 77 By calculation, the identity to be proved is at once reduced to the relation. MEESSLERI EE for 0 5 we obtain, exactly as on p. 485, the relation stated, namely % E Ima 20d rh a 1 n-t1 as a necessary condition for the E -summability of the series Ja, . To determine further what the condition E,-lim¢ = 0 implies as regards the order of magnitude of the terms @,, we deduce from ar =r [(G) a +o +7) a] the expression for the g,’s in terms of the a4 '’s n n 3 oom 43a = (aad == (0) whence, as a,’ — 0, it at once follows, by 48, 5, that 20 or ua, ={37. If we carry out the corresponding calculation for s, and s/', we simi- larly find that s, = 0(3"). Summing up, we therefore have Theorem 1. The four conditions . : LD dla E,-limga, —=0, E,-lim 420 tn @,%=0(3%) ond ss =0(3") n =), are necessary in order that the series 2a,, with partial sums s,, may be summable E,. A comparison of this theorem with the theorems 271 and 283 shows that the range of the E,-process is considerably more extensive than those of the C- and A -processes; the E -process is a good deal more powerful than these. The question, however, which in the case of the C- and A- processes led to the theorems 274 and 287, here reveals what may be described as a loss of semsitiveness in the E,- summation process, as compared with the C- and A4-processes. We in fact have Theorem 2. If the series 2a,, with partial sums s,, is sum- ~mable E, to the value s, so that the numbers @ =L[G et Gutta] and if, besides this, we have (B) a =o), yo then the series Xa, is convergent with the sum s - (0-E — K-theorem). § 63. The E-process. 518 Proof. As in the case of the proof of 374, we form the dif- ference ’ 1 % 4p 4 ST 50" ola 2 J), — San)» and we split up the expression on the right hand side into three parts: 1, +7,4-7,.- 7, is 10 denote the pat from y=0 10 v=, 7, the part from v = 3#n to v = 4 n, and 7, the remaining part in the middle. In virtue of (B) there certainly exists — roughly estimated — a con- stant K; such that |s,| < K, Vn. Hence there also exists a constant K such that 15, = Sau| SK Vn for every n, provided 0 < v < 4%. Hence |T,] and |T;| are both hE Bn) <5 Now for every integer & > 1, we have A ENE 23 e(2) 1: ¢, =2(ab+a'’b,_,+--+a'b))— (a) bp-a+ "+ an_s18)). Since by hypothesis one at least of the two series Xa,’ and Xb ' is absolutely convergent, Cauchy's product, X(a,/b + ---+ a,b), of these two series is convergent and = A-B, by 188. From the last obtained expression for ¢ ’, the convergence of X'¢,’ follows at once, and for its sum C we obtain C=24A8B— AB=A8, g-e d% Finally, we shall examine the question of the relation between the C- and IE -processes. IL is very easy to show, in the first in. stance, that the processes fulfil the compatibility condition 263, III; i. e. that we have Theorem 4. If a series is summable C; and also summable E,, the 293. two processes give it the same value. Proof. If (¢,’) is the C,-transformation and (s/) the E,-trans- formation of (s,), both these sequences are convergent, by hypothesis: say ¢,'—c, s,'—s’. Since both processes satisfy the permanence condition, the C,-transformation of (s,’) also converges to s”: Est n+1 and the E,-transformation of (c,’) converges to ¢’: Las +(e] With the abbreviated notation, these two relations are CE (s)—7% and E C,(s)—¢. But, as was pointed out on p. 511, these two sequences are identical, 50 that §¢ must be equal to ¢/, gq. e. 4d. We have already seen (p. 510), from the example of the geometric | series Xz", that the E, - process is considerably more powerful than the C,-process. In fact, we cannot sum the geometric series by the latter anywhere outside the unit circle, while the FE, -process enables us to sum it at every point of the circle |z-}1| < 2. But this must not be interpreted to mean that the range of action of the E,-process completely includes that of the C,-process, much less those of all C,-processes. On the contrary, it is easy to give an instance of a se- 51 The form of this proof suggests that in certain cases it will be con- venient to introduce the concept of absolute summability: A series 3a, will be said to be absolutely summable E, if Xa,/, its E,- transformation, converges absolutely. 294. 295. 516 : Chapter XIII. Divergent series. quence (s,) which is limitable C,, but not limitable E,. The sequence s)=0, 1, 0, 0, 2, a, 0, 0, Q; 3, 0, rie is of this type, where s,. =» and every other s = 0 (i. e. for every index # which is not a perfect square). ” This sequence is limitable C;, with the value 3 For the largest So+S1+: 15a ’ v RT are obviously attained for n=1? and the least for # =—=»2—1. The latter = ga , the jormer values of the arithmetic means (r+1)» Sis 1. 1 = TY both of which — 5; i.e. C5) 5 If the same sequence were also limitable E;, we should therefore require to have E, (s, ) — 3 but, for n =»?, the (2 n)® term of the E,- transformation is 1 2n 2n 2n 1 /2a\ay— — ered pL a Ys, I I Coun Ve The expression on the right hand side tends to > by 219 3, so JT > that for all sufficiently large »’s the terms remain > > > and E, (s,) cannot — 3. The sequence (s,) is therefore nos limitable E;. We may accordingly state: Theorem 5. Of the two ranges, that of the C,-process and that of the E,-process, neither contains the other entirely. There are series which can be summed by the C,-process, but not by the E,- process, and conversely. This circumstance raises the further question: which series, summ- able C,, can be summed by the E -process? Little is as yet known on this subject, and we shall content ourselves with mentioning the follow- ing theorem: Theorem 6. If 2a, is summable C,, with So =Siroce 15, 1X rl sel then Za, is also summable E,. (o-C,— E,-theorem *.) 2 The statement remains the same when higher orders of both processes are considered. : 88 As on p. 486, footnote 43, we have called the above theorem the - 0-C, — E,-theorem. The corresponding O-theorem does not hold, as has already been shown by the example on theorem 5, where the arithmetic mean is actually Feo § 63. The E-process. 517 Proof. Writing s, — s = ¢,, the sequence (o,) is limitable C; with the value 0. Put Oo=0y re 00 c!: n41 Tm by the hypotheses, we then have not only ¢,”— 0, but also Vno,'—O0. Now we have the following general inequality, due to Abel: If 1% 40, 4-10, es mT yp=0,1,.... 7) the hs en arithmetic means; if, further, 7 is a number greater than all the (#n-}1) quantities |g’|, for »=0,1,...,%, and 7, a number greater than all the quantities |g,’ |, for £<» 3, to the numbers ¢,, 6, intro- duced above, and taking ¢,= Sa »=0,1,2,...,%, we can choose for 4 the greatest integer < 225 Le 2 -[ 7]: and we may similarly take m = 5]: We then obtain we [()oo + (a+ + (a) Sut ea) + 5m). This inequality will hold a fortiors if we assume 7 to be greater than all the quantities |o,| and 7, greater than all the quantities |o,’ with » >%k. Now, by 219, 3, the greatest term te) of the values / 4: . woe . ) satisfies the limiting relation Vu (un 1 on £) TE VT 7a on on we then have also 27 n = a am wh <7,Vn<32 Vp. so that () is certainly << 1 from some stage on. From this stage 8 In fact, oy = vt 1)o), — vo,_4 and frente bie Cts =0, $00, = Zeta lo—mi) = + + >. v=0 r=p v=m Noting that a. iD is negative in the first and second sums and positive in the third, it follows that | 2 eros | Shy ry mm tn (mt tt tga 4a), »=0 ‘whence the above relation follows directly. 518 Chapter XIII. Divergent series. Since 7, Vp was to tend to O, since, further, $ and m tend to Joo as un does, and at the same time £0) — (, it follows that when n— 00 7 [(5) + (Donte (3) oa] 0 ie e's: ged Exercises on Chapter XIII. 200. With the help of example 119 (p. 270), prove the fact mentioned on p. 461, namely that wo [L101 imo 1 z>+0 p=1 n® 2 What summation process related to the A-process might be deduced from this? Define it and indicate some of its properties. 201. Is the condition ; a, +2a,L...+ na, Gy -m (BE2H TAY Lg, given in 273, substantially equivalent a) to Cp+1-lim(na,)=0, Db) to C,-lim (Bit bra) = 0? n 202. With reference to the relations (¥) in the proof of 284, show that in general 4-lims, =s always involves 4 C;-lims, =s, i. e. - sh (1—2) Ts,z”—>s always involves (1— =. 2 2" —>s, n+ k\ Sas (for z —1—0). 203. Show similarly that B-lims, =s always involves B C,-lims, =0, i.e. e=% > Hi x nk yah Ey for x —» + 00. 204. Are the conclusions mentioned in 202 and 203 reversible, e. g. does the relation (1—z) > ED (1-2) Xs,a"—>s? z et E— 205. The series = 1 Put z=cosg +ising, Sani the real and imaginary parts and write down the trigonometrical series summed in this way, as well as their respective values. E. gz, a" — s imply, conversely, that Yon is summable C, for |z|=1, z+ +1. 1+ 2cosz+ 3 cos 2u+ 4 cos Bator m op ——— 4 sin® =~ 2 cosz +2 cos Der Sed Saliva dail etc, 4sin? Exercises on Chapter XIII. 519 206. If (a,) is a positive monotone null sequence, and if we put aya, +a, tr ta,=10b,, A is summable C, to the sum =o 3 DVa,. the series 207. If we write 1 + : + z dee > = h,, it follows from the preceding exercise that the C,-sum By ly ly By me mt Tor 8, and similarly that the C,-sum log2 —1log3+1log4d—+--.- =z, 208. If Xa, is convergent or summable C; with the sum s, the follow- ing series is always convergent with the sum s: > a,+2a,+---+na, “t 2 n(n-+1) = 209. If Ya, is known to be summable C, and X# |a, |? is convergent, then Xa, is itself convergent. 210. Prove the following extensions of Frobenius’ theorem (p. 491): If Za. is summable C, to the sum s, then for z— 1 (within the angle) n=1 aD RD N02. ad Na, 5s n=1 n=0 and in general, for every fixed integer p =>1, own - Out ors, n=0 But Ya,z" does not necessarily tend to s, as may be shown by the example 0 > (— 1)2z™ n=0 for real x—1—0. (Hint: The maximum of ¢/—¢", and the value of ¢ for which it is attained, both — 41 from the left as #» increases.) 211. i a 7c! series Sua, is not summable C,, but Za,2” is con- vergent for 0< x <1, then, for every ¢ >0, a 8 >0 can be assigned le] such that the sum of > a,x” lies between » —¢ and p+ & for every z in n=0 1—d1-—0 215. Give a general proof of the commutability of the E,- and C,-processes. 216. Deduce the 0-E, — K-theorem (what is its statement?) by induction from the o0-E, — K-theorem. 520 Chapter X1V. Euler's summation formula and asymptotic expansions. Chapter XIV. Euler's summation formula and asymptotic expansions. § 64. [Euler’s summation formula. A. The summation formula. The range of action of all the summation processes with which we became acquainted in the last chapter was limited. It is only when the terms a, of 2'a,, the divergent series under consideration, do not increase too rapidly as zn increases that we can sum the series. Thus in the case of the B-process, the most powerful of the processes . . . iii a which are useful in practice, it is necessary that Si should be n —— * a convergent everywhere, i. e. that I ol or 23 a,| should tend to zero. Hence the B-process cannot be used e.g. for the series =m tia 0) 31h 4] be sorb fe Pl hones n=0 Series like this one, and even more rapidly divergent series, occurred, however, in early investigations of the most varied kind. In order to deal conclusively with them by the methods used hitherto, we should have to introduce still more powerful processes, such as the B, -process. We can make no progress in this way, however, since the processes naturally become more and more complicated. At a fairly early stage in the development of the subject other methods were indicated, which in certain cases lead more conveniently to results useful both in theory and in practice. In the case of the numerical evaluation of the sum of an alternating series 2(—1)"q,, in which the q's constitute a positive monotone null sequence, we observed (see pp. 250 and 251) that the remainder 7, always has the same sign as the first term neglected, and, moreover, that it is less than this term in absolute value. Thus in the calculation of the partial sums we need only continue until the terms have decreased below the required degree of accuracy. A somewhat similar state of affairs exists in the case of the series x2 xn gri=1l—u-Lo +t) Fy z>0, since the terms z likewise decrease monotonely when n > xz. We can therefore write i x2 nz" wy ntl e Fr=1~ go 55 frie =D (1) frm ro for every m > x, where J stands for a value between 0 and 1, but § 64. Euler's summation formula. — A. The summation formula. 521 is otherwise undetermined. It is impossible in practice, however, actually to calculate ¢=* from this formula when z is large, for e.g. 8000 As 1000! is a number with 2568 digits (for the calculation see below, p. 531), the term under consideration is greater than 10'3!, so that the evalu- ation of the sum of the series cannot be carried out in practice. From the theoretical point of view, on the other hand, the series fulfils all requirements, since its terms, which (for large values of x) at first in- crease very rapidly, nevertheless end by decreasing to zero, and that for every value of x. Hence any degree of accuracy whatever can be obtained in theory. The circumstances are exactly the reverse, if we know that the value of a function f(x) is represented by the formula? when x = 1000, the thousandth term is equal to oh FB) =1= 2 +2 — ho (C0 2 pg ED O0 [rf do = Sv (f= fr) == Got fi e+ F) +0 + Ds ry=1v—-1 Since » = [#] +1 in the »th interval of integration, we may write n fotfite+h=0+0f— (+f @ de. Now by integration by parts we immediately obtain n nhy=[2f'@dz+ [1(@) de, so that n n fot hit +ha=Jr@de +1, + [@— [2] - Df @) d=, 2 With regard to the summation formula cf. footnote 4, p. 523. — The phenomenon described above was first noticed by Euler (Commentarii Acad. sc. Imp. Petropolitanae, Vol. 11 (year 1739), p. 116, 1750); A. M. Legendre gave the name of semi-convergent series to series which exhibit this phenomenon. This name has survived to the present time, especially in astronomical literature, but nowadays it is being superseded by the term “asymptotic series”, which was introduced by H. Poincaré on account of another property of such series. 3 In the subsequent remarks all the quantities are to be real. ~., § 64. Euler's summation formula. — A. The summation formula. 523 or, re-writing in a form more convenient for later use, i : : 296. fit fit fom [FO dwt 3 Got 1 + (2 — [2] = 2) F @ de. This in fact is Euler's summation formula in its simplest form 4. .It gives a closed expression for the difference between the sum ff, 4 +f, n and the corresponding integral J f(@) dz. 0 We shall denote the function which appears in the last integrand by P, (2): 1 : - P,@y=u—[o]— 5, , This is essentially the same function as the one which we met with in one of the first examples of Fourier expansions (see pp. 351, 375), It is periodic, with period 1, and for every non-integral value of z we have sindumy. P@E=-3 A simple Ci to begin with will illustrate the importance of this formula. If f(z)=-—— = we obtain, by replacing # by nn —1, Py (2) , z2 1 1 1 1 af n—1 2 We may substitute the latter integral for Jars: a ia dx, since P, (x 1-1) = P, (2). As P(x) is bounded in 22> 1, the tater obviously converges when n» — 00, and we find that 0 : 1 1 21 Pp lim (145400 —logn) = ro alee n>» 9 1 4 The formula in its general form 298 originated with Euler, who men- tioned it in passing in the Commentarii Acad. Petrop., Vol. 6 (years 1732-3, published 1738) and illustrated it by a few examples. In Vol. 8 (year 1736, published 1741) he gives a proof of the formula. C.Maclaurin uses the formula in several places in A Treatise of Fluxions (Edinburgh 1742), and seems to have discovered it independently. The formula became well-known, especially through Euler's Institutiones calculi differentialis, in the fifth chapter of which it is proved and illustrated by examples. For long it was known as Maclauvin's formula, or the Euler-Maclaurin formula; it is only recently that Euler's un- doubted priority has been established. The remainder — which is most essential — was first added by S. D. Poisson (v. Mémoires Acad. scienc. Inst. France, Vol. 6, year 1823, published 1827). The particularly simple proof given in the text is due to W. Wirtinger (Acta mathe- matica, Vol. 26, p. 255, 1902). An up-to-date, detailed, and expanded treatment is to be found in N.E. Nor- lund’s Differenzenrechnung, Berlin 1924, especially in chapters II—V. 297. 594 Chapter XIV. Euler's summation formula and asymptotic expansions. We already know that this limit exists, from 128, 2. Now we have a new proof of this fact, and in addition we have an expression in the form of an integral for Euler's constant C, by means of which we can evaluate the con- stant numerically. If we now wish to proceed to further developments of the formula n n 1 / (O) fthitot f= [1@dr trot f+ [PE Ee 0 0 obtained above, integration by parts is plainly suggested as a suitable expedient. In order to be in a position to carry it out, we must first assume that f(x) has continuous derivatives of all the orders which occur in what follows; then we have to select an indefinite integral of P, (x), and an integral of the latter, and so on. By suitable choice of the constants of integration the further calculations are greatly simpli- fied. We shall follow Wirtinger® and set w a 2cos2nnz BE =+2 my Then P,’ (x)= P, (x), for every non-integral value of z, and P,(0} es Bl = Moreover, P, (x) is continuous throughout, and has the period 1. We now proceed to set 2 Asin2naz B= Yr whence we have P)’(x)= P, (x) for every value of z, P, (0)=0, and in general Il ZL 2 CO82N TT Ps (0) =(—1) ee =1 (2nnx) n 1 = 2s 2nax Ps eet "lL x Snag, 2241 ( ) ( ) ath @nnx)tl (a) Then, for 1=1, 2,..., all these functions are throughout continuous and continuously differentiable ¢, and have the period 1; and we have ; Pi.y (2) = P(2), BD (o=tg- St, oy Zant. CH’ Pir41(0y=0 for 2, A=1,2,...(cf. 136). As is immediately obvious from the proof, for the interval 0 <2 <1 we have to deal with rational integral func- tions, and in fact it follows at once from the last formula, combined 8 Cf. the last footnote. ® Except that P, (x) is not differentiable for integral values of z. § 64. Euler's summation formula. — A. The summation formula. 525 with the fact that P,(%) — 2 — = in 0 1. Hence, for every 2 >> 0, provided only that the derivatives of f(x r) involved exist and are gontinucus, we can write: B98. 7447, + stp rosning At) : B, B, + Cl Em I oS (FES fmt) Bi where we put 2% +1 TE np IEE my dn =R, for short. This is Euler's summation formula. Remarks. 1. Since in the last integration by parts, namely n n @K) 7. _ EE @k+1 = 7:7 duv=e|Pu 7 LJ Pera ‘de, the integrated part vanishes, on account of the fact that P, 2k+1 (n) = Py. 4(0), we may also write _ fru red (@)dz 0 for the remainder in the summation formula. 2. If we put F(a+xh)=[ (x), the formula takes the somewhat more general form, in which F(a)+F(a+h) +--+ F(a+nh) forms the left hand side. The formula may therefore be used for the summation of any equidistant values of a function. 8. With suitable provisos, it is permissible to let # — 00 in the sum- mation formula. According as Xf, converges or diverges, we then obtain an expression for the sum of the series or for the growth of its partial sums. The statement is different (on the right hand side) for every value of &. 4. If we let 2 — 00, Ry may tend to 0. We should then have an infinite series on the right hand side, into which the sum on the left hand side is transformed. This case actually occurs very seldom, however, since, as we are aware (v. p. 2387, footnote), Bernoulli's numbers increase very rapidly. 9 For 2 =0 the terms involving Bernoulli's numbers of course do not occur. § 64. Fors summation formula. — B. Important applications. 527 The series 2a [2% n. = ls T will turn out divergent for almost all | (2 oa ! 1 the Sn f@) which occur in applications, no matter what » may be. Thus the formula suggests a summation process for a certain type of divergent series. Very valuable results are obtained by making # or % increase in a suitable way (see B, 3, 4, 5). 5. Provided that the differences (f® i have the same sign n 0 ? the series just discussed is an alternating series, since the signs of the num- bers By, are alternating. We shall see that, in spite of the divergence, the above-mentioned evaluation of the remainder of the alternating series remains valid. 6. The formula will be useful only in the cases where, for a suitable value of k, Rj is small enough to give the desired degree of accuracy. At first sight, we have, according to 297 (a), only the inequality 2 Zl 4 == eter To [2 (x) | = (2 w)~ = nk = (2 zt at our disposal for the estimation of Ry, for k = 2: but, as we see subsequently, the inequality also holds for k=1, and by 136 it can be put in the more |B precise form | Py (x) | < 2 for even values of k&. 7. Notice, finally, that the formula is not only a summation formula, but is equally an integration Sormala, as it can shvionsly be used for the approxi mate evaluation of the integral fre as also. B. Important applications. 1. It is obvious that the most favourable results are obtained 299. when the higher derivatives of f(x) are very small, and especially when they vanish. We therefore first choose f(x) = x?, where $ is an integer > 1, and we have n P2713 1... tof = [oP dot La? + pur eee, 0 Here the series on the right hand side is to be broken off at the last positive power of zn, for [Pia YP ude vanishes not only when oo (z)=0, but also (by 297 b) when f® (2) is identically equal to a non-vanishing constant. Thus by transferring 7? to the right hand side we have 124-27 fof bln = 1) =o lt Cm er 1 Pmt 4), 528 Chapter XIV. Euler’s summation formula and asymptotic expansions or — since there is no constant term appearing inside the brackets on the right hand side? — PEt det) 1 1 1 spl B) i= nr, 2. The sums dealt with above can be obtained in quite a differ- ent way. If we imagine that each term of the sum 1+ et + et} vuu J gn—DE » is expanded in powers of #, the coefficient of 7 is obviously 274. +n — 1)”. On the other hand, if we use symbolic notation (cf. 105,5), the first sum is equal to 1 + nt (n+B)t Bt Le pL. e —e t = t gHn= ef—-1 1 . Hence we immediately obtain the expression - {(n + Byrtt ie pzit) for the coefficient of = 3. If we put fa n=1, we obtain i 5 (e® +1)=""- + Ze Her—1 LE oR aa 0 or 1 8 alas EB e*—1 2 a, 2k +2 : 2 T 2 en” 5 a Pypiq(2)e dz. 0 Since we can immediately prove, by 298, 6, that the remainder tends 2 to zero in this case, provided only that [&¢| < 2x, we have, for these values of «, © 9 2v B 2! 2 ne Le 2 which 18 the expansion stated in 105 Similarly, by putting f(x) = cosa«x, n= 1, we obtain the expan sion 115 for 5 Cot 5 10 Thus in the notation of p. 525, footnote 8, we have 1 17 +27 1 Te ey, y= yn) § 64. Euler's summation formula. — B. Important applications. 529 4. If we put f@)= 11s we have, by replacing # by (n — 1), 1 1 i 1 B 1 B 1 145 +b =logn +5 +o +3 (1-5) +7 (1-33) +o (1) — +1) 24ti a Since here we may let n — 00, just as on p. 526 above, we obtain the following refined expression for Euler's constant: B, 7 Code 2 rt tb. a — ent 225, xX In this case the remainder Sp does not decrease to 0 as k in- creases; and the series niu i k diverges rapidly, — so rapidly that even the corresponding power series 3 ou ge diverges everywhere; for, by 136, 202k | Bil = = 7: where 1 << m2. Nevertheless, we can evaluate C very accurately by means of the above expression (cf. Rem. 4). If we take e.g. k=238, we have, in the first instance, dz. : 1. at tia P, (x) sod plo tl &) C=vrp—mim 7 z® 1 If we take only the part of the integral from z=1 to x=4, the absolute value of the error (v. Rem. 6) is 71 fp 4-7! ly 2a) mwa 1 Hence 4 1459 P, (z) elf Ls de + 755, where Iyl» exists. Its value is obtained as follows: by (*) we have 2log(2-4-...2n)=2nlog 2 -} 2log n! =2nlog2 + (2n-+1logn —2n+-27y, =2n-+1)log2n—2n—1loz2}2yp, and 3 log(2n +1)! = (22+ 4)log (22 +1) = (20 Dol 5,040 By subtraction 2.4:6-...22 1 1 1 bog v3 31 ord — @n+1)log (1 = ry —glog(2n +1) 1 1—~10g2-|2y pay = § 64. Euler's summation formula. — B. Important applications. 531 If we now transfer the term log (2m +1) to the left hand side and let #— co, we know, from Wallis’ product (p. 384), that log/F = —1+1—log2 +27 —7, so that y = log Ven. Hence we now have 00 n If we multiply by M, the modulus of the Briggian logarithms (pp. 256—7), and denote the latter logarithms by Log, we have Logn!= (n+) Log n —n M+ Log Tau 1210] n This gives, e. g. for n = 1000, 00 Log 1000! = 3001-5 — 434:29448 . . . + 039908... — M £9 dz. 1000 Since x gr x 1000 ZX 1000 1000 ou emi = 12000 T (2% 1000 — 10000’ it follows that Log 1000! = 2567-6046. . . with an error << 10—4 in absolute value, so that 1000! is a number with 2568 digits, which begins with the figures 402... . Just as in the previous example, we can now put our result in a more refined form by means of integration by parts. Since p Pres 0), 5 Piri la) [220-2080 2000 iy, n after 2 k steps we obtain 1 — By 1, By 1 logn!=(n-+4+ logn—n+logVeat im ot —+ vive By, 1 [ut os 0 eee. § Se eet een i ! me . + OI=1% nik—1 (2F) ZR T dx n As here the remainder (for fixed k) is less than a certain constant divided by n?¥, we can also write the result in the form By 1 By. 1 4, n\? ENT RET nl=\— Vane ’ 532 Chapter XIV. Euler’s summation formula and asymptotic expansions. in which the A,’s always (i.e. for every fixed ) form a bounded sequence, since in fact |4,] remains less than (2k —1)! The result 4 in either form is usually known as Stirling's formula.lt 6. If we take the somewhat more general form f(z) = log (y + x), where y > 0, Euler's formula for 2 = 0 gives, to begin with, : log y 4log (+04 log (y+) = (y+ w)logly +n) —n — ylog y +5 (log (y +n) + log) +] 2a We can obtain the gamma-function (v. p. 385 and pp. 439—40) from this expression as follows: subtract this equation from the one obtained in the last example, namely from logn! 4 logn¥= (n+) logn — n+ ylogn +10g Van — [2 da, n in which we have added log n¥ to both sides, and we obtain nlnY L I TR GT = (v— 5) 103 y—(y+n+7)log™s wa (@) (7, @ +logV2n— IE = [Ea dz. If # — co, this relation becomes 1 Pn log I'(y) = (v — 5 )logy — y+ log 2 = (219, 0 By integrating this expression by parts 2 k times, as we have already done several times, (or by at once using Euler's formula for any value of k), we deduce the following generalized Stirling's formulal?: log I' (y) = bp iey y+logVen B, 1 B,,. 1 + tae + Eran Ee i Pa (x) § w+ etl az — (2K)! 1t J. Stirling, Methodus differentialis, London 1730, p. 135. But the fact that the constant y is log Vem does not appear till later. 13 Stirling (loc. cit.) gives the formula for the sum log z+ log (x +a) + log (x+2a)+ +4 log (x + na). § 64. Euler's summation formula. — B. Important applications. 533 7. We now put =r where x > 0 and s is arbitrary. + As we have already dealt with the cases s =1, —1, —2,..., and the case s = 0 is trivial, we shall consider s as being different from any of these values. If we again replace » by (n — 1), Euler's formula now gives tf tot Leedplie ta 0] Ce HEE Se ig =e 2 a Pojpyq @ ; — (2&+1)! Lie oars. 1 If s>1 we can let n— co, and we obtain the following remarkable expression for Riemann’s {-function (cf. pp. 345, 445—6, and 492): (0 =p tHE) RCE s+2k Prg+1(x) mer (52 fF ei a 1 Since the right hand side has a meaning for s > — 2%, s 41, and since k can take any positive integral value whatever, we immediately infer from the above — the details of the proof belong to the theory of complex functions — that 1 s—1 £(s) — is an integral transcendental function (cf. p. 493, footnote 54). Further this expression gives the values 1 and for s = — p (p a positive integer), if we suppose that 2k > 4: tnt e deen tHE Here the series terminates of itself, and we can write p+1 21 pl ras B+) B+ (3) Bt at pl Bye, Fit +B) Pl’ where the last step follows from the fact (v. 106) that A BY — BP l=. 18# 534 Chapter XIV. Euler's summation formula and asymptotic expansions, 8. Particularly interesting results are obtained by choosing ta, e>0,z4+0; [(0)=0, az 1 qx so that f(x) is a function which is differentiable as often as we please for real values of x <= 0. Since by 114,4 f@)= a+ biuret for sufficiently small values of x, f(x) is also differentiable as often as we please at x = 0, and we have f® (0) = Zar, (p an integer > 1). Euler's summation formula now gives 1 1 1 1 1 1 n eee tek ip te) 1 1 =p %% Fak. = [loz = +5], +5) 2 B Ba = Bt 2k - oh (ee on? of rly p00 Mm ie om oly i +P 2141 (0) F040 (@) do. The expression in the square bracket has the value 1 1 n 1 =—log{l — ene) — log ut = — loge. Transfer the terms log and to the left hand side and let n — 00: then pd ces] — log n— C, and Fm) — 5 Hence, if we can show that the derivative f®(n)— 0 for every index 1 >1, and that the infinite integral converges, we obtain 1 1 Soy By? Bf rg = Clog) 1 991% Tai hal! 01 2 B} ~ rary” [ru (@) fE¥+ 1 (2) da. The assumptions which we have just made with reference to the integral and the derivatives are easily proved; for, since a 1 ei m 3 m du (¥—1)m (2 1)” {2-171 ? § 64. Euler's summation formula. — B, Important applications. 535 f(x) is obviously of the form C C. 0) = yo e Ci +1 1) ps 2h {22 = 1) for 2 => 1, where the Cs are definite constants. The required con- ~ clusions follow at a glance; for the convergence of the integral which - stands for the remainder follows from the fact that |F® (x)| remains < —T when x — 00, if K is a suitable constant. A closer approximation to the value of the integral is somewhat more troublesome to obtain. We may proceed as follows. By 2189, fx) = > 2 — = eint + 4vial’ so that, if we put uz 1 1 {= rrr = ax —2vmi + cx+2rmai ? we have WP @) = (— 17 ple? (—— tb), (¢z—2vmi)?Ptl ' (qz42vai)?tl If ext 2vai=reciv : 1 2v x (p22 2,98, -1 7= (022+ 47%2%)2; p=1tan =, so that i 2 cos ((p +1) tan-1 277) o WY (0) = (= 17 ple? bf (ee? 22+ 4 v2 72) 2 and hence freesuisy = Sm @)] < rr EH Bo IL r=1(¢? 2+ 42a) HT Taking into account the fact that (@* 2? 4 49222) > (2a? + 492 2%) (2 v0) 2k 2 (022? 4 42%) (2v a)h, it follows that a?x®4 4 a? ma k 00 [rer @)| < 2.254110 . 2 : 2k+1 =@k +1): By | ores 536 Chapter XIV. Euler's summation formula and asymptotic expansions. Hence | Pays 1 (@) F259 (2) dor) 0 SEE 1B rit tf dx @n2tl | fet dat" 0 Now the value of the integral is = so that, remembering that 1 | Bar | (2x)2% = 2-24)! by 136, we may finally write @k+1 Boy got Py (0) FEF (2) de = 3 py (22 + Drs 0 in which the absolute value of § is less than 1. The expression ob- tained in this way for the sum of the scries = 1 na __ 1° (¢ > 0), n=1 € i. e. of the Lambert series (v. § 58, C) oo yr T=»? with y SE e=2, n=1 for positive values of y <1, is the more favourable the smaller « is, i. e. the closer y is to +1 — i. e. in the very circumstances in which the series under consideration converges very slowly. C. The evaluation of remainders. The evaluation of the remainder in the last example was fairly troublesome, and in previous examples we passed over this question altogether. The question arises whether it is possible to make a general statement as to the magnitude of the remainder in Euler's summation formula. We shall see that this is actually possible in very general cases, in such a way that, as already indicated in A, the remainder is of the same sign as, but smaller in absolute value than, the first term neglected, — i. e. the term which would appear in the summation formula, if we replaced 2 by 2+ 1. This will, moreover, always be the case if f(x) has a constant sign for x > 0 and if f(x) and all its derivatives tend monotonely to 0 as x tends to -- oo. In order to prove this, we must examine the graph of the function y= P, (x), k > 2, in the interval 0 < x <1, somewhat more closely. We assert that the graph is of the type represented in Fig. 13; 1, 2, 3, 4, according as k leaves the remainder 1, 2, 3, or 0, when divided by 4. § 64. Euler's summation formula. — C. The evaluation of remainders. 537 | i ¢ s 4 ° 4 1 2. e, o. Fig. 13. More precisely, we assert that the functions with odd suffixes have exactly three zeros of the first order at 0, 1, 1, but those with even suffixes exactly two zeros of the first order within the interval, and, moreover, that the functions have the signs shown in the graphs. More shortly: Py;(z) is of the type of the curve (— 1)*-1cos2mua and Py;4q(%) is of the type of the curve (— 1)*~1sin2anw. These statements are proved directly for the suffixes 2, 3, 4, by using the methods which follow. We may therefore assume that the assertions are proved up to Py; (z), 4 > 2, inclusive. It is immediately obvious, by 297, that Py; (x) vanishes for 2 = 0, 3, 1, and also that Poip1(1 — x) = — Payyq(®), so that Py; (#) is symmetrical with respect to the point x — > y ==), Thus if Py;44 (x) has another zero, it must have two more at least, i. e. five in all, and P,, (x) must have at least four zeros (by Rolle’s theorem, or by § 19, theorem 8), which is contrary to hypothesis. The sign of Pp (®) in 0 <2 < 3 is the same as that of Pj; (0) = P,,(0), that is, the same as that of B,,; i. e. the sign is given by (— 1)*-1. Since Pj;49(®) = Psj11(%), Pait2(®) has only one stationary value in 0< <1 namely at = — 1. Its value Py;.9 (3) must have the opposite sign to Py; (0), for otherwise P,;,q(x) would have a con- stant sign in 0 R, and represents the func- tion F(x), the series is obviously an asymptotic representation of F(z) in this case also. Thus examples of asymptotic representation can be obtained from any convergent power series. 3. The question whether a function F(z) possesses an asymptotic re- presentation, and what the values of the coefficients are, is immediately settled in theory by the fact that the successive limiting values (for z — 4 00) F(zx)—> a,, (F (x) —a,) x —> a, , (Fer—a—2) a®—>a,, must exist. In fact, however, the decision can seldom be made in this way; but these simple considerations show that any function can have only one asymptotic expansion, 4, On the other hand, for f(z)=¢~%, > 0, all the a,’'s are zero, since xke—%—>0 § 65. Asymptotic series. 543 for every integral 2 => 0, when x — oc. Thus 9-0 er" ~O0t ste. aa a result which shows that different functions may have the same asymptotic expansion. Thus, if F(z) has an asymptotic representation, e. g. Bl) tem? BlnY-Lae 2 (5>0), ..., have the same asymptotic representation. It was for this reason that we could not inter that the two functions mentioned in 300,2 were identical. 5. Geometrically speaking, the curves y=a +2442 and y=F() have contact of at least the nn order at infinity; and the contact becomes closer as » increases. 6. For applications it is advantageous to extend the definition by writing F@) ~f@)+8@) (a+ 24224.) if Fo) —f) Tam and f(z) and g(x) are any two functions which are defined for sufficiently large values of z, and if, further, g(x) never vanishes. Some of the examples workea out in § 64, B, may be regarded as giving the asymptotic expansions, in this sense, of the functions involved, for we may now write ~at 2p 1 1 1 a) 1% ator be ajlogn SCL pm — bf — ry inne] oo! 1 m=, B, 1 1 b) tog nl ~ (n+) tog n—n) + log Y2a + 5+ ot gyrate 1 5, By 1 1 c) tog (I'(@) ~ ((v— 3) log x — x) + log Za + 15: Tefal Eo 1 1 ood d) thie pla lonr ols ok) +1] 2(; 1 5, s+2\1 . 18 ely de Nt 3 Yi Z 1 Bed B21 SSE sr a Eloget Cot 3-20. BET Sn 18 gs 1; for s=1 it becomes the expansion in a). ) 544 Chapter XIV. Euler's summation formula and asymptotic expansions. Calculations with asymptotic series. The concept introduced by definition 301 owes its chief useful: ness to the fact that in many respects we can make calculations with asymptotic series just as we do with convergent series. It is immediately obvious that from F(x) ~ a, iL 22 ot and b b CRB Fath... there results the expansion b + pb F(@)+ BG) ~way+ p+ ETE £0tlh yp where ¢ and § are any constants. It is almost as easy to see that the product of the functions also possesses an asymptotic expansion, and that F@)G@~c +2424. if, as in the case of convergent series, 2b, ab, to +a,b, is set equal to c¢,. For, by hypothesis, we may write (for fixed z), a Ay a, ang F@)=ay+2 + = Toe Set b b y, but 1 GB iat Bop ie Simp 20, if by e=¢ (x) and # = 7 (x) we denote functions which tend to 0 as Z— -F 00. But in this case we have [F@)6@— (+ 2+2 + +2)" —an+be Ln a, (bp+n)+ayby 1+ +--+ (a,+¢€)b; i Et bn x ah and this obviously tends to 0 as xz — - oo. Repeated application of this simple result gives 30 2. Theorem 1. If cach of the functions F, (x), F,(®), ..., F, (x) possesses an asymptotic representation, and if g(zy, 2y, -+.,2,) is a poly- nomial, or — if we anticipate what ay follows — any rational function whatever, of the variables z;, zy, ..., z,, then the function Oe litho § 65. Asymptotic series. 545 also possesses am asymptotic representation; and this is calculated exactly as if all the expansions were convergent series, provided only that the denominator of the rational function does not vanish for 3 = (). Further, the following theorem also holds: Theorem 2. If g(2) = aty +, 2+ --- + «0, 2" + ++ is a power series with positive radius », if F(x) possesses the asymptotic representation Fa)~ay+2 +24, and if |a,| R, say, we have to set F(z) =a,-}-f, and — assuming only that | a, R if the expansion for f(x) does so: but even when this is not the case, the quantities a have quite definite values, as they are rational integral functions of a finite number of the quantities @¢,. We must now substitute these expansions in (*) and arrange the result formally ; ; . : ; 1 (i. e. just as if the series (**) were convergent) in powers of x. 546 Chapter XIV. Euler's summation formula and asymptotic expansions. Thus we obtain an expansion of the form 4, Lo La 2 2 4-.. where the cocfficients are given go i=l, CC di=Pa), A= foast Fin, A4,=0.0,+0e%+t: +58". We have to show that 2 is an asymptotic expansion of @ (x), that is, that the difference : OE) — (4, +2 +2 + +2), when multiplied by 2", tends to O for fixed #=20,1,2,..., as x —00.. Now if ¢ =¢, (7), &, =¢, (x), ... all denote functions which tend to 0 as x — 00, it follows from 0 and (**) that B@)=f,+ By (24 +t 8) bo 4B, (4 2) feted. + Bp at 4] 4 . En+1 Hence, since f»*+! may be put equal to ey 3 D@)— (Ag+ 2+ +2) = Lhe, + Pay +o + foe] $222 Bors thane hired] zn and our assertion follows at a glance, for the expression in the last square bracket tends to f3, ,, as x — -}- 00, and the (finitely numerous) g's tend to O. From this, it follows as a particular case, provided only that a, + 0, that n+1 1 1 | a’—a,a, 1 le tnd iL seal aud Bla) "wa, =" afz a,’ hg 2 Hence we “may” divide by asymptotic expansions with non-vanishing constant terms; this completes the proof of theorem 1. Similarly — and, moreover, without any restrictions — an asymp- totic expansion may be used as the exponent of e: Saline lies 103 a, Bel pd In particular, we may write, by 299, 4, 1 1 139 1 nl (3 J Vean1 +90 Ts ea +]. Term-by-term integration and differentiation are also valid with suitable provisos. We have § 65. Asymptotic series. 547 Theorem 8. If F(x)~ a, > he pe +--- and if F(x) is con- tinuous for x > x, [oe] y@ = (FO) — a — 2) a oT ebedy Pilon ole SHEE nEe If F(x) has a continuous derivative, and if F'(x) does possess an asymptotic expansion, the latter is Ble is 20 sais, SE st Prooi. Since #2 (7 {D— a, — 2) — a, as {— 4-00, the integral which defines the function w(x) always exists for a > x,. Further, we may set Fl te Sk er iii {n 2 1, fixed), t tn+1 {ni1? where &(f)—0 as {— 4 co. Hence Guy (t pa) —2—... es [a Now if & (2) denotes the maximum value of |e ( z) in x...00, E(x)—0 after multiplication by z" it likewise tends to 0. Now if the derivative F’(x), which is continuous for x > z,, possesses an asymptotic expansion / b, by ZA bytt A Lobes we have F(z) — [Fw di 1-C, — [(s AL 5) dt + [Eo 0 pp 2) dik C, — bya +b Joga + C, — [(F'0) — 0, — 4) a, z where C,, C, are constants. By what we have just proved, and be cause a function defines its asymptotic expansion uniquely, it follows that b,=0b, = 0 and that b, = — (mn — 1)a,_, for un > 2. The function Fe) =e "sin() ~ 0+ 5 +o + exemplifies the fact that F’(z) need not possess an asymptotic ex- pansion, even when F(z) does. 548 Chapter XIV. Euler's summation formula and asymptotic expansions. Theorems 1—3 lay the foundation for Poincaré's very fruitful applications of asymptotic series to the solution of differential equations? A detailed account lies outside the plan of this book, however, and we must content ourselves by giving an example of this application of asymptotic series in the following section (in A, 1) § 66. Special cases of asymptotic expansions. The use of asymptotic expansions raises two main questions: first there is the question whether the function under consideration possesses an asymptotic expansion at all, and how it is to be found in a given case (the expansion problem); on the other hand, there is the question how the function, or rather, a function, is to be found, which is represented by a given asymptotic expansion (the summation problem). In the case of both questions, the answers available in the present state of knowledge are not completely satisfactory as yet, for although they are very numerous and in part of remarkably wide range, they are somewhat isolated and lack methodical and funda- mental connections. This section will therefore consist rather of a collection of representative examples than of a satisfactory solution of the two problems. A. Examples of the expansion problem. 1. From the theoretical point of view, the expansion of given functions was thoroughly dealt with in the note 301, 3; but it is only seldom that the required determinations of limits can all be carried out. The method also fails if lim F(x) does not exist, i.e. if only an T>®o asymptotic expansion in the more general sense mentioned in 301, 6 can be considered. It is only when f(x) and g(x), the functions in- volved, have been found that we can proceed as in 301, 3. 2. We have learned that asymptotic series very frequently arise from Euler's summation formula (v. 301, 6): but there it is not so much a case of expanding given functions as that by special choice of the function f(x) in the summation formula we are often led to valuable expansions. 3. As we have already emphasized, hitherto perhaps the most important application of asymptotic expansions is Poincaré's use of them in the theory of differential equations?®. The simple fundamental 19 A very clear account of the contents of Poincaré’s paper, including all the essential points, is given by E. Borel in his “Legons sur les séries diver- gentes” (v. 266). 2 § 66. Special cases ot asymptotic expansions. 549 idea is this: suppose we know that a function y = F (x) satisfies a differential equation of the nth order Die, v, ¥; ores ¥) = 0, where @ denotes a rational function of the variables involved. Now if we know that y — F (x) and its first # derivatives all possess asymptotic representations, the expansions for y’, ¥”, ..., y™ follow, by 302, Theorem 3, from the first one, ya, ttt Lo, If we substitute these expansions in the differential equation, in accord- ~ ance with 302, Theorem 1, we must obtain an expansion which stands for 0, all the coefficients of which must therefore vanish (by 301, 3, 4). From the equations obtained in this way, together with the initial condi- tions, the coefficients and hence the expression for F(x) are in general found. Thus e. g. the function @® =! y=F@) =e" a, x which is defined for x > 0, has for its derivative @ —-Z ~t y =F @ =e [Sar—er —y 1, z that is, it satisfies the differential equation 1 Yr rheem= 0 for x > 0. It may be proved directly — but we cannot give the details here — that this equation has only one solution y such that y and y’ exist for x > 2, > 0 and have an asymptotic representation. If we accordingly set a, ay ’ a, 2a, ; Visa, =t Ax mh mbes, so that y VERT aE Tl we have the equations 4G=0, a,=1, a,=—a,,"+s HB 1 =—N0,; +4n whence it follows that n 2=0 i a=, Gm—1,..., a y= nln We therefore find that 1 1 21 3! Tsar mtamt me 550 Chapter XIV. Euler's summation formula and asymptotic expansions. 4. The function in the previous example can be asymptotically expanded by another method, which is frequently applicable. If we put {=u -}z, we have F@) =| oT du. 0 Here, by Cauchy's observation (v. 300, 1), we can put 1 1 u u?2 u™ nid unt emt hr Beant) O—5e Od «1, for all positive values of x and #. It follows that 1 1 21 n! +14 (+1)! Ee =o bn rheor pf pe bet 9, a OF, <1. Thus not only have we again found the expansion in the last example, but we have at the same time obtained a fresh example for the ob- servations made in § 64, C 2°. 5. If f(u) is a function which is defined for # > 0 and is posi- tive there, and if the integrals Fr ur ldy = (— le, 0 exist for every integral # > 1, we similarly obtain the asymptotic ex- pansion a Fla tor tof pees x for the function © F(z) | rn du. 0 Moreover, the partial sums of this series represent F(z), except for an error which is less in absolute value than the first term neglected and is of the same sign as the latter. Expansions of this kind have been investigated especially by Th. J. Stielijes?'. (For further part culars, see below, B, p. 554). © p30 20 The function ¢”? F (x) -| 2 x i e : dv : , which becomes — | —— with the logv 0 t transformation ¢~°=1v, is known as the Logarithmic-integral function. 21 Stieltjes, Th. J.: Recherches sur les fractions continues, Annales de la Fac. des Sciences de Toulouse, Vols. 8 and 9, 1894 and 1895. § 66. Special cases of asymptotic expansions. 551 6. Certain methods requiring the more advanced resources of the theory of functions date back to Laplace, but have recently been ex- tended by E. W. Barnes ??, H. Burkhardt®3, O. Perron??, and G. Faber?® We cannot go into details, but must content ourselves with the following remarks. Barnes gas the asymptotic expansions of many integral ET: (£0, —1, — 2, ee and similar func- tions. Besides the expansions we have met with and others, O. Perron obtains as examples the asymptotic expansion in terms of n of certain integrals which occur in the theory of Keplerian motion, such as functions, e.g. 3 i (¢—esing) : am= [0 a, (0 < eX 1, wan integer), 1 — cost -JT + C (n) = [ ent—sinii gg, From our point of view it is noteworthy that in these examples the ” : 4 1 terms of the expansion do not proceed by integral powers of 7) but by fractional powers. Thus the expansion of C(n) is of the form Cn) ~—t i 5 = Beet 2 =» : This suggests another extension of the definition $01, 6, which, however, we shall not discuss. Numerous additional examples of asymptotic expansions of this kind, in particular those of trigonometrical integrals occurring in physical and astronomical investigations, are to be found in the article by H. Burk- hardt: “Uber trigonometrische Reihen und Integrale”, in the Enzyklo- padie der mathematischen Wissenschaften, Vol. II, 1, pp. 815—1354. 7. An expansion, which was first given by L. Fejér 2%, and was subsequently treated in detail by O. Perron ?’, is of a more specialized nature; its object is to deduce an asymptotic representation for the x . . . Of =r coefficients of the expansion in power series of ¢ 1~%, or, more gener- 22 Barnes, E. W.: The Asymptotic Expansions of Integral Functions defined by Taylor’s Series, Phil. Trans. Roy. Soc., A, 206, pp. 249—297. 1906. 28 Buykhavdt, H.: Uber Funktionen grofier Zahlen, Sitzungsber. d. Bayr. Akad. d. Wissensch., pp. 1—11. 1914. 24 Peyyom, O.: Uber die niherungsweise Berechnung von Funktionen grofier Zahlen, Sitzungsber. d. Bayr. Akad. d. Wissensch., pp. 191—219. 1917. 20 Faber, G.: Abschitzung von Funktionen grofier Zahlen, Sitzungsber, d. Bayr. Akad. d. Wissensch., pp. 285—304. 1922. 26 Fejér, L.: in a paper in Hungarian. 1909. 27 Peyyon, O.: Uber das infinitire Verhalten der Koeffizienten einer ge- wissen Potenzreihe, Archiv d. Math. u. Phys. (3), Vol. 22, pp. 829—340. 1914. 552 Chapter XIV. Euler’s summation formula and asymptotic expansions. ally, of e1-2)¢, where 9 > 0. We at once find that x Tre] 2 Y 2 e = SrlerT)=1+tartar =e where the coefficients c, have the values 1 ou w~ (C0) It is not known whether for these coefficients, regarded as functions of n, there exists an asymptotic expansion in the sense used hitherto or in the extended sense of paragraph 6. If it does exist, however, it is of the form ‘ YaevEn 1 fn ec [> T J since the ¢’s are asymptotically equal, in the sense of 40, 5, to the above function of #. Regarding the following term, we only know ve 1 that it is at least of the order ts, 8. Finally, we draw attention to the fact that the asymptotic re- presentation of certain functions forms the subject of many profound investigations in the analytical theory of numbers. In fact, our examples 301, 6, a,b, and d, belong to this class, for the functions expanded have a meaning only for integral values of the variable in the first instance. Just to indicate the nature of such expansions, we give a few more examples, without proof: a) If 7(n) denotes the number of divisors of #, Drier. (1) 2 310 ~ log anlC = A) Loos, where C is Euler's constant ?®. Regarding the next term 3%, all that is known is that it is lower in degree than #~% but not lower than 7%. b) If o(n) denotes the sum of the Ses of n, o(1)+0©2)+-- tol) = vty n #8 An elementary proof of the less complete result logcy ~ 2Vant--- is given by K. Knopp and J. Schur: Elementarer Beweis einiger asymptotischer Formeln der additiven Zahlentheorie, Math. Zeitschr., Vol. 24, p. 559. 1925. 29 Lejeune-Divichlet, P. G.: Uber die Bestimmung der mittleren Werte in der Zahlentheorie (1849), Werke, Vol. II, pp. 49—66. 30 Hardy, G. H.: On Dirichlet’s Divisor Problem, Proc. Lond. Math. Soc. (2), 15, pp. 1—15. 1915. § 66. Special cases of asymptotic expansions. 558 c) If (nn) denotes the number of numbers less than # and prime to it, Dtg@+---+om _ 3 e)+9@ ig EE n d) If 7(n) denotes the number of primes not greater than #, aia In all these and in many similar cases, it is not known — cf, the corresponding remark in paragraph 7 — whether a complete asymptotic ~ expansion exists. Hence the relation which we have written down only means that the difference of the right and left hand sides is of smaller order, as regards #, than the last term on the right hand side. n 2 (n) ~ log n e) If p(n) denotes the number of different ways in which # may be partitioned into a sum of (equal or unequal) positive integers ®., p(n) ~ dn 38 2 > In this particularly difficult case G. H. Hardy and S. Ramanujan succeeded by means of very profound investigations in continuing the expansion to terms of the order ——. Vn B. Examples of the summation problem. Here we have to deal with the converse question, that of finding a function F(x) whose asymptotic expansion is the given series 2 2g Gyr ok = 2h fa es which diverges everywhere 3. The answers to this question are still more isolated and lacking in generality than those of the previous division. When the function F(z) is found, it has some claim to be re garded as the “sum” of the divergent series Sox in the sense of § 59, since it becomes more and more closely related to the partial sum of the series. This is the case only to a very limited extent, however, since, as we have already emphasized, the function F(z) 3 e.g. p(4)=05, since 4 admits of the five partitions: 4, 841, 2+ 2, 24141, and 1414141. 32 Hardy, G. H., and S. Ramanujan: Asymptotic Formulae in Combinatory Analysis, Proc. Lond. Math. Soc. (2), Vol. 17, pp. 75—115. 1917. 3 In the case where the series converges, the required function is in fact defined by the series. 554 Chapter XIV. Euler's summation formula and asypmtotic expansions. depends on the way in which it was found and is not defined uniquely by the series. Thus the question how far F(x) behaves like the “sum” of the series can only be investigated in each particular case a posteriori. 1. The most important advance in this direction was made by Stieltjes 3%. We saw above (v. A, 5), that a function given in the form o-f iu 0 possesses the asymptotic expansion z= in which {) (1 =a" du, © #=1,28y. 0 Conversely, if we are given the expansion nl Su i. e. a perfectly arbitrary sequence a,, 4,, 45, ..., and if we can discover a positive function f(u) defined in x > 0, for which the integral in (¥) has the given values g,, —a,, 4;, ..., for n=1, 2, 3, ..., the function u FPl)= [re du 0 formed from this function f(x) will be a solution of the given summa- tion problem, by A, 5. The problem of finding, given a,, a function f(x) which satisfies the set of equations (*), is now called Stieltjes’ problem of moments. Stieltjes gives the necessary and sufficient condi tions for it to be capable of solution, and, in particular, for the existence of just one solution, with very general assumptions. In particular, if f(u), and hence F(x), is uniquely determined by the problem of moments l — such a series Sr “is called a Stieltjes series for short — we are quite justified in claiming F (x) as “the sum” of the divergent series } =. Lack of space prevents us from entering into closer details of these very comprehensive investigations. An account which includes everything essential is given by E. Borel in his “Lecons sur les séries divergentes”, which we have repeatedly referred to. As an example, suppose we are given the series 1 ! (1) ET aT ET ETT 3 Loc. cit. (footnotes 16, 21), and also in his memoir, Sur la réduction en fraction continue d'une série procédant suivant les puissances descendantes d'une variable, Annales de la Fac. Scienc. Toulouse, Vol. 3, H. 1-17. 1889. § 66. Special cases of asymptotic expansions. 555 The statement of the problem of moments is @0 fra urtdu=(n—1), n=1 2... 0 which obviously possesses the solution f(u)=e~%. In this case we can prove by Stieltjes’ theory that the above is the only solution. Hence in Ron Fix) — [Za 0 we have not only found a function whose asymptotic expansion is the given series, but, in the sense of § 59, we can regard F(z) as the sum of the (everywhere) divergent series (f) obtained by the Stieltjes process — and we shall call this sum the Stieltjes sum for short. 2. The appeal to the ‘theory of differential equaiions is just as useful in the summation problem as in the expansion problem (v. A, 3). Frequently we can write down the differential equation which is formally satisfied by a given series. If we succeed in solving this differential equation directly, among the solutions there may be a func- tion whose asymptotic expansion is the original series. Asa rule, however, matters are not as described above, nor as in A, 3, but the differential equation itself is the primary problem. The latter can be solved for mally by means of an asymptotic series, as was indicated in A, 3. It is only when we succeed in summing the series directly that we can hope to obtain a solution of the differential equation in this way. Otherwise we must try to deduce the properties of the solution from the asymptotic expansion. Poincaré's researches ®, which were extended later, espe- cially by A. Kneser and J. Horn ®?, deal with this problem, which lies outside the scope of this book. 3. In Stieltjes’ process the coefficients a, were recovered, so Ay a”’ to speak, from the given series x by replacing a, by n=1 fra Whee du 0 85 Thus for x =1 we obtain the value 0 s= [en £7 = 0500517... 142 0 @D for the Stieltjes sum of the divergent series 3 (— 1)" n! This series had already n=0 been studied by Euler, Lacroix, and Laguerve. Laguerre’s work formed the starting-point of Stieltjes’ investigations. % Poincaré, loc. cit. (footnote 17). 37 A comprehensive account is given by J. Horn: Gewd6hnliche Ditferential- gleichungen beliebiger Ordnung, Leipzig 1905. 556 Chapter XIV. Euler's summation formula ana asymptotic expansions. and hence the series by Veins, +o) du. 0 In place of the series = there now appears the very simple geo metrical series; the factor f(u) occurs instead. The solution of the problem of moments is necessary in order to determine f(u), and this is usually very difficult. Now we can make the process more elastic by not recovering the coefficients completely, but only so far as on the one hand to make the solution of the problem of moments sufficiently simple, but on the other hand not to let the new series become too complicated. In other words, we put Op ax \ 1 B= Se (2) x and choose the factors ¢, so that the problem of moments © n ¢,= J flu) u" du 0 can be solved, and the power series fe (2) On NT represents a known function. Thus we obtain a connection between this result and Borel's summation process in § 59, by putting ¢, = nl. We then take f(u) =e~% and if PREVI) is a simple function, 0 F (2) — [eva (®) du may be a solution of the given summation problem 38, Here we cannot discuss the details of the assumptions under which this method leads 3 We can recognize the connection with Borel's summation process as follows. The function 2 oo xh y=e > Sn 7 n=0 : which was introduced on p. 472 in connection with Borel's process, has for its derivative oo zh L. — Y=2"2 au = n=0 : G 7 : Ri dy Thus if we set Sits & (t), as in the text above, we have v' =e" % & (x), n=—~n * § 66. Special cases of asymptotic expansions. 557 to the desired result. We shall conclude with a few examples of this method of summation: in these, however, the question whether the function found is really represented by the series must remain un- settled, or be verified a posteriori, since we have not proved any general theorems. a) For the scries 1 1! ! 3! SRT Em Ty which we have already discussed in 1, we have a,—(— ne? (wn —1)!, so that @ (£) == log (x = 2), and accordingly F(z) — [em log (1+ =) du. 0 An easy transformation (integration by parts) shows that the function is identical with that discussed in 1. b) If we are given the asymptotic series 1 1-3 185...@n=1) legytoiy—cb-rd i- lf eieth u —k we have ® (2) = (1 +=) E so that a 0 oy oo P= Ve [du 2°Vz [emt au. u 0 Zz Va Thus we have an asymptotic expansion for the function : 2 1 orl 1 Gl)= dt =ge? (—gm t+) which is of special importance in the calculus of probabilities. so that z y=a,+ [et & (1) dt. . 0 Hence if the B-sum of Xa, exists, it is given by @ s=a,+ [ eto (Yd, 0 — an expression which, with suitable assumptions, can be transformed into ow J e~!® (f) dt by integration by parts. This corresponds to the value of F(1) 0 deduced in the text. 19 558 Chapter XIV. Euler’s summation formula and asymptotic expansions. c) If we are given the somewhat more general series 1 1 n 1 1— tomel alt De—f 101) e(@41)(et+n—1)— Lae, u w\=2 we have @ (%) == (1 +2) , so that rE ee? 1 Fo at Fo) =z | du="Lgeer [ort i¢ “di, J ta) z% d) For the series 1 2! 4! Ee TT we have @ (=) == fan—1 (2), so that © on i F(x) — [ee tan-1 (=) du = Shen du. 0 0 It ums 1s regarded as the Stielfjes sum of the given divergent series, we obtain e. g. the value Ss 0 for the sum of the series oe 20 LY BY Sein du = 0:6214 . Exercises on Chapter XIV. 217. Using the symbolic notation which was introduced in 105 and 297, show that the following relations hold: a) B+1+2)"— B+)’ =n], for n=1, b) F(B+1+42)—F(B+a)={(a), SY FOLIO) Lf =F (BL 140) F (541), where in b) and c¢) F (2) denotes a polynomial and F’ (2) =f (2). 218. By considering special cases of Ex. 217, b) and c), deduce the following (symbolic) relationships: 9 (Brg) -(B-g) =O", ez, b) iL )(B+D(B+)...B+m=nl, @Z1), ¢) B+1)?PB?—(B-1)?B?=0, 1>1, 16-2436 1... 4 (1 Ils =U iz sin +875 = 59% + 32). * Exercises on Chapter XIV. 559 b) Generalize this result and prove the following statements: If 2 er symbolically, we have, in the first instance, 428) i-@aprit 2138, tts Dri an Samy, OF Ty 2! n! Ca n+1 ntl and (C+1*4-C*=0 for n=1, so that 1 1 1 G=1, Co=Ctn C,=0, Cro C,=0, Csr) 8.9/9 Using these numbers, we have (again symbolically) 1002 4-30 ite vn ra? = {= DPC HTH mP — CP. 220. a) Prove that (1-2-3)? — (2:83:42 +++ 4 (~ D1 (n n+ 1) n+ 2)? RI 1 . (» + 9nd + 28nt-33n3-LTn2— Cn, b) Referring to Ex. 219,b), generalize this result and deduce a formula “for the sum FO=F@+fB)—+--- +=)" fm), . where f(x) denotes a polynomial. 221. a) What are the explicit formulae for a? + (a+b)? +(a+2b)P 4... + (a+ nb)? and —@+b0)?4+(@a+20)P—+... + (—1)" (a+ nb)? p being a positive integer? b) As in 296 and 298, deduce a formula for for fut Fort sorb 1% Fons 222. Referring to the integral test 176, show that the following is true: Let Xa, be a divergent series of positive terms, which, however, form a bounded sequence (a,), and let s, denote the partial sums of the series; further, let f(x) be a positive continuous function which decreases monotonely. Then Saf) amd [ras w=i 1 either both converge or both diverge. Prove this, and, in the light of the theorem 175, 2, the corollary to 176, and the remarks preliminary to 296, investigate the connection between the partial sums of the series and the corre- sponding parts of the integral. 223. We have oe : nw : mrs [te = Jim [0 Yet TET +r 560 Chapter XIV. Euler's summation formula and asymptotic expansions. Here Euler expanded term by term in a geometrical series, and collected in powers of x, and, for the particular case x =1, obtained LL glall PNT -n" 22 nt a eT) Tr 6 n? = 1 i] 5 1 7 1 BE 3a Tee 25a 6 oT TT Deduce this expansion by applying Euler's summation formula to f(z) = a . 224. a) As in 299, 3 and 4, deduce the expansion in power series for the function a by applying Euler's summation formula. b) Apply Euler's summation formula to the polynomials Py (x), and hence evaluate the integrals 1 / 2, (2): P, (x) dx 0 3 (m and n positive integers). [= (e+ 5) P, lx) dn 0 . c) In Euler's summation formula put f@=acloge, allogz, =ztlogx, a(logz)? ... and investigate the relationships thus obtained (v. Ex. 234 below). d) In Euler's summation formula put f(z) =e **7% where «>0, 1>0, and hence prove the relations ee Zr r(; wa” 3.1 gt ive [Rees 144 and 1 3 1 a =31(3)= lidar. JO hail, 1s 5 —n 225. Euler's summation formula 298 can of course be used equally well for the evaluation of integrals as for the evaluation of sums. Show in this way that 4 [arom .. 0 (v. Ex. 232 below). 226. a) The sum 12241... 0 has the value 7-485470 ... for #=1000, and the value 14392726... for #=1000000. Prove this, first assuming that C is known, and then without assuming a knowledge of C. b) Prove that n! has the value 10498578, 2.8949, ., 3 for n =105, and the value 1075363708, gona... for n= 108. 561 Exercises on Chapter XIV c) Prove that r(a+3) has the value 102%.19723.., for x =103, and the value 105365705, sonny... for a= 101. d) Without assuming a knowledge of the value of #2, evaluate 3 ] 1 1 1 EEL hs for #=10 and find the limit of this sum as n—»00. (We obtain 0-10416683. 0-10516633 ...). e) Using d), show that a = 164493406... . t) Show that x > — 1-20205690.. and that 2 Sl any... n=1,7% ae has the value Prove that fats eb g) 72 $= 5 1998.54014 .. has the value 521.57582... for nn =10°%, aD h) Prove that. 3’ il for ai Tm 227. By 299, 6, u (x) = log I' (2) (2-3) logz+4z- logy2x = [7 On, 0 for > 0. Hence deduce Gudermann’s Series for pu (x), namely 1 )-1]. 2 I i= n-4—) lo ( es # (@) 3 |(o+ +3) gilts 228. For the evaluation of { (s) for s=1+4 6, where § is small and pos- itive, we have the formula 1 J 1 (6+1) @4+D(E+2)(E4 BY hse Ty te 0° 720-10" mea]. Complete the series in square brackets, and show that in its case also the error is always less in absolute value than, and of the same sign as, the first L(A+0)=1+ ot he term neglected. 562 Chapter XIV. Euler's summation formula and asymptotic expansions. 229, Prove that the function a raat 2 , considered in 299, 9, does es—1"z' 2 not possess the property which is necessary in order that theorem 300 may be applicable, namely that the function itself and all its derivatives should de- crease monotonely to 0 in 0 0 and for n=0,1,2,..., and are such that pr {2 e=v pum (2 ) du =0 z> +o \? 0 for fixed », the given series actually is the asymptotic representation of F(z). 234. Prove the following relationships, stated by Glaisher: n (n+1) IY 2) 22.8%, on? yom Dive to, (ct. Ex. 224, c), A el in 1-550 - (Aw —B)i ere 1.44.97, ,,.. (Bn Dirt 2:53.88... (Bn 1)ir-t b) A, (An 1), ~ 4, (3n)2nt1, where 4,, 4,, 4,, are definite constants. The latter have the following values: dual 1/71 1 1 4, = 27% 50 €Xp [s(Fe+ga-gu+—] 2 1 1 4,~exo[- 3 Z(1-gutge—+eo)]s = 2 2s Zom J 53 pls: Ragas -]s 1 7 17 Das where C is Euler's constant and s; denotes n 235. Obtain the asymptotic expansion for the function (== 1 1 Flos =f eg 42° Sut dee and prove that the coefficient a, of r is given by 236. The numbers considered in Ex. 112 occur in the expansion in power series of 2 (GES) 0 $° =o0 me XN E1y, Prove that the following asymptotic evaluation holds for these numbers: logg, =n [108 7 ah = , where ¢g,—>0, or 8 _ i n = rt where 7,—0 564 Bibliography. 10. 11. 12. = 183. 14. 15. 16. 17. 19. . Fabry, E.: Théorie des séries a termes constants. Paris 1910. 21. 22. 23. Bibliography. (This includes some fundamental papers, comprehensive accounts, and textbooks.) . Newton, I,: De analysi per aequationes numero terminorum infinitas. Lon- don 1711 (written in 1669). . Wallis, John: Treatise of algebra both historical and practical, with some additional treatises. London 1685. . Bernoulls, James: Propositiones arithmeticae de seriebus infinitis earumque summa finita, with four additions. Basle 1689—1704. . Euler, L.: Introductio in analysin infinitorum. Lausanne 1748. . Buler, L.: Institutiones calculi differentialis cum ejus usu in analysi infini- torum ac doctrina serierum. Berlin 1755. . Euler, L.: Institutiones calculi integralis. St. Petersburg 1763—69. . Gauss, C. F.: Disquisitiones generales circa seriem infinitam wf, aletD-F@E+D) Aan, Gottingen 1812. . Cauchy, A. L.: Cours d’analyse de l’école polytechnique. Part I. Analyse algébrique. Paris 1821. . Abel, N. H.: Untersuchungen iiber die Reihe 1420+ 20 Vp... Journal fiir die reine und angewandte Mathematik, Vol. 1, pp. 311 to 339. 1826. du Bois-Reymond, P.: Eine neue Theorie der Konvergenz und Divergenz von Reihen mit positiven Gliedern. Journal fiir die reine und ange- wandte Mathematik, Vol. 76, pp. 61—91. 1873. Pringsheim, A.: Allgemeine Theorie der Divergenz und Konvergenz von Reihen mit positiven Gliedern. Mathematische Annalen, Vol. 35, pp. 297—394. 1890. Pringsheim, A.: Irrationalzahlen und Konvergenz unendlicher Prozesse. Enzyklopddie der mathematischen Wissenschaften, Vol. I, 1, 3. Leipzig 1899. Borel, E.: Lecons sur les séries a termes positifs. Paris 1902. Runge, C.: Theorie und Praxis der Reihen. Leipzig 1904. Stolz, O., and A. Gmeiner: Einleitung in die Funktionentheorie. Leipzig 1905. Pringsheim, A. and J. Molk: Algorithmes illimités de nombres réels. En- cyclopédie des Sciences Mathématiques, Vol. I, 1, 4. Leipzig 1907. Bromwich, T. J. I'A.: An introduction to the theory of infinite series. London 1908: 2 ed. 1926. . Pringshetm, A. and G. Faber: Algebraische Analysis. Enzyklopiddie der mathematischen Wissenschaften, Vol. II, C. 1. Leipzig 1909. Nielsen, N.: Lehrbuch der unendlichen Reihen. Leipzig 1909. Pringsheim, A., G. Faber, and J. Molk: Analyse algébrique, Encyclopédie des Sciences Mathématiques, Vol. II, 2, 7. Leipzig 1911. - Stolz, 0. and A. Gmeiner: Theoretische Arithmetik, Vol. II. 22d edition. Leipzig 1915. Pringsheim, A.: Vorlesungen iiber Zahlen- und Funktionenlehre, Vol. I, 2 and 3. Leipzig, 1916 and 1921. Name and Subject Index. The references are to pages. Abel, N. H. 122, 127, 211, 281, 290 seq., 299, 313, 314, 321, 424 seqq., 459, 467, 564. Abel's partial summation 313, 397, 522. — limit theorem 177. — extension of 406. — convergence test 314. Abscissa of convergence 441. Absolute convergence of series 136seqq., 396. — of products 222. Absolute value 7, 390. Adams, J. C. 183, 256. Addition 5, 29, 31. — term by term 46, 68, 134. Addition theorem for the exponential function 191. — for the binomial coefficients 209. — for the trigonometrical functions 199, 415. Aggregate, closed 7. — ordered 5. a’ Alembert, J. 459. “Almost all” 63. Alterations, finite number of, for se- quences 45, 68, 93. — for series 130, 476. Alternating series 131, 250, 263 seq., 316, 520. Ames, L. D. 245. Amplitude 390. Analytic functions 401—2. — series of 429. Andersen, A. F. 489. Approach within an angle 404. Approximation 63, 231. Archimedes 6, 104. Area 169. Arithmetic, fundamental laws of, 5. — means 70, 460. Arrangement by squares, by diagonals, Avzela, S. 344. Associative law 6. — for series 132. Asymptotically equal 66. — proportional 66, 249. Asymptotic series (expansion, repre- sentation) 540 seqq. Averaged comparison 464, 466. Axiom, Cantor-Dedekind 25, 32. Barnes, E. W. 531. Bernoulli, James and John 17, 63, 184, 238, 457, 525seq., 564. Bernoulli's inequality 17. Bernoulls, Nicolaus 324. Bevnoulli’s numbers 183, 203—4, 237, 479, 524 seqq. — polynomials 525, 536 seqq. Bertrand, J. 282. Bieberbach, L. 478. Binary fraction 37. Binomial series 127, 423-8. — theorem 48, 190. Bécher, M. 350. Bohr, H. 493. du Bois-Reymond, P. 66, 85, 94, 301, 304, 305, 353, 355, 379, 564. du Bois-Reymond's test 315, 348. Bolzano, B. 85, 89, 394. Bolzano-Weierstrass theorem 89, 394. Bonnet, 0. 282. Boormann, J. M. 195. Borel, E. 320, Yt seqq., 477, 548, 554, 556, 564. Bound 14, 158. — upper, lower 94, 159. Bounded functions 158. — sequences 14, 42, 78. Breaking off decimals 249, Briggs, H. 56, 257. Bromwich, T. J. 477, 564. Brouncker, W. 104. Burkhardt, H. 353, 375, 551. 190, 208-11, Cahen, E. 290, 441. Cajovi, F. 322. Cantor, G. 25, 32, 66, 355. — A 11. Cantor-Dedekind axiom 25, 32. Cavmichael, R. D. 477. Catalan, E. 247. ; Cauchy, A.L. 17, 70, 85, 94, 104, 113, 117, 136, 138, 146, 147, 148, 154, 186, 196, 219, 294, 408, 459, 539, 550, 564. 566 Cauchy's inequality 408. — limit theorem 70. — product 147, 179, 489, 514. Cauchy-Toeplitz limit theorem 72, Centre of a power series 157. Cesare, E. 292, B18, 322, 466. Chapman, S. 477. Characteristic of a logarithm 56. Circle of convergence 402. Circular functions 57: see also Trigono- metrical functions. Closed aggregate 7. — expressions for sums of series 232 to 240. — interval 18, 162. Commutative law 5, 6, 10. — for products 227. — for series 138. Comparison tests of the first and second kinds 113seq., 274 seq. Completeness of the system of real numbers 32. Complex numbers: see Numbers. Condensation test, Cauchy’s, 120, 297. Conditionally convergent 140, 226 seq. Conditions F 464. Continued fractions 105. Continuity 161-2, 171, 174, 404. — of power series 174, 177. — of the straight line 25. — uniform 162. Convergence 62, 76seq. — absolute 136 seq., 222. — conditional, unconditional 140, 227. — of products 218, 222 — of series 101. — uniform 326 seq., 428 seq. Convergence, abscissa of, 441. — circle of, 402. — criteria of: see Convergence tests also Main criterion. — general remarks on theory of, 298 to 305. — half-plane of, 441. — interval of, 153, 327. — radius of, 151. — rapidity of, 251, 262, 279, 332. — region of, 153. — systematization of theory of, 305 to 311. — tests for Fourier series 361, 364—372. — for sequences 76—86. — for series 110—120, 124, 282—290. — forseries of complex terms 396—401. — for series of monotonely diminishing terms 120—6, 294, 296. — for series of positive terms 116, 117. — for uniform convergence 3328. Convergent sequences: see Sequences. Cosine 199seq., 384, 414 seq. 3 Index. Cotangent 202seq., 417 seq. Curves of approximation 329, 330. Decimal fractions 116: see Radix frac- tions. — section 22—23, 49. Dedebind, B. 1, 25, 82, 40, — section 40. Dedekind’s test 315, 348. Dense 11. Diagonals, arrangement by 88. Difference 29, 343. Difference-sequence 85. Differentiability 163. — of a power series 174—35. — right band, left hand 163. Differentiation 163—4. — logarithmic 382. — term by term 175, 342. Dini, U. 227, 282," 290, 293, 511, 344. Dini’s rule 367—S8, 371. Dirichlet, G. Lejeune- 138, 329, 347, 356, 875, 552. Dirichlet's integral 356seq., 359. — rule 865, 871. Dirichlet series 317, 441 seq. Divichlet's test 315, 347. Disjunctive criterion 118, 308, 309. Distributive law 6, 135, 146 seq. Divergence 63, 101, 160, 391. — definite 64, 101, 160, 391. — indefinite 65, 101, 160. — proper 65. Divergent sequences 457 seqq. — series 457seqq. Division 6, 30. — term by term 46, 69. — of power series 180 seqq. Divisors, number of 446, 451, 552. — sum of 451, 552. Doeisch, G. 478. Double series, theorem on, 428. — analogy for products, 437—438. Duhamel, J. M. C. 285. e 80, 1948. — calculation of, 251. Eisenstein, G. 180. Eliot, EB. 814. e-neighbourhood 18. Equality 26—7. Equivalence theorem of Knopp and Schnee 481. Evrmakoff’s test 216seqq., 312. Error 63. — evaluation of: remainders. Euclid 6, 18, 67. Eudoxus, postulate of, 10, 26, 33. — theorem of, 6. see Evaluation of Index. Euler, L.. 1,80, 104, 182, 193, 204, 211, 228, 238, 243, 244, 262, 853, 375, 384, 385, 413, 415, 439, 445, 457seqq., 468seq., 509seqq., 520 seqq., 541seqq., 564. Euler's constant 225, 228, 271, 524, 529 seq., 541, 543, 552, 5635. — @-function 451, 553. — formulae 353, 415. — numbers 239. — transformation of series 244—6. Evaluation, numerical, 247—260. — of e 251. — of logarithms 198, 254—7. — of & 252. — of roots 257—3S8. — of trigonometrical functions 258—9. Evaluation of remainders 250, 527, 536—540. — more accurate, 259. Even functions 173. Everywhere convergent 153. Exhaustions, method of, 67. Expansion of elementary functions in partial fractions 205—S8, 239, 377, 419, 528. — of infinite products 437. — problem for asymptotic 548 seqq. Exponential function and series 148, 191—18, 411—4. Expressions for real numbers 230. — for sums of series 230—273. — for sums of series, closed, 232—240. Extension 10, 32. — condition 463. series Faber, G. 483, 551, 564. Fabry, E. 267, 564. Faculty series 446 seq. Fejér, L. 494, 497, 551. Fejér's integral 495. — theorem 494—5. Fibonacci’s sequence 13, 270, 452. Finite number 13, 16. — — of alterations: see Alterations. Fourier, J. B. 852, 375. — coefficients, constants 354, 361, 362. — series 350seqq., 493 seqq. — Riemann’s theorem on, 363. Frobenius, G. 184, 491. Frullani 375. Fully monotone 263, 264, 305. Fundamental laws of arithmetic 5, 31. — of order 5, 28. Function 158, 403. — interval of definition, limit, oscilla- tion, upper and lower bounds of, 158—9. 567 Functions, analytic, 401 seq. — arbitrary, 351. — cyclometrical, 2183—5, 421 seq. — elementary, 189 seq. — elementary analytic, 410 seq. — even, odd, 173. — integral, 411. — of a complex variable 403 seq. — of a real variable 158seq. — rational, 189seq., 410 seq. — regular 408. — trigonometrical 198seq., 258, 414 seq. — sequences of, 327seq., 429. Glamma-function 225—6, 885, 440, 532. Gaps in the system of rational num- bers 3seqq. Gauss, C.F. 1, 113, 177, 238, 289, 564. Gibbs’ phenomenon 379, 497. Glaisher, J. W. L. 180, 563. Gmeiner, J. A. 399, 564. Goldbach 458. Goniometry 415. Grandi, G. 133. Graphical representation 18, 891. Gregory, J. 63, 214. Gronwall, T. H. 879. Gudeymann, C. 561. Hadamard, J. 154, 299, 301, 314. Hagen, J. 182. Hahn, H. 2, 305. Half-plane of convergence 441. Hanstedt, B. 180. ” Hardy, G.H. 318, 822, 407, 444, 477 seq., 486, 487, 552, 553. Hausdorff, F. 418, 483. Heymann, J. 131. Hilbert, D. 10. History of infinite series 104. Hobson, E. W. 350. Holder, O. 465, 491. Holmboe 459. Horn, J. 555. Hypergeometric series 289. Tdentically equal 14. Identity theorem for power series 172. Induction, law of, 6. Inequalities 7. Inequality of nests 28. Infinite number 13. — series: see Series. Infinitely small 17. Innermost point 21, 394. Integrability in Riemann's sense 166. Integral 165 seq. ¢ — improper 169—170. Integral test 294, 522. 568 Integration by parts 168—9. — term by term 176, 341. Interval 18. — of convergence 153, 327. — of definition 158. Intervals, nest of, 19, 394. Inverse-sine function 215, 421 seq. Inverse-tangent function 214, 422seq. Improper integral: see Integral. Isomorphous 9. Jacobi, C. G. J. 439. Jacobsthal, E. 245, 263. Jensen, J.L. W.V. 72, 74, 441. Jones, W. 253. Jordan, C. 14. Keplerian motion 551. Kneser, A. 555. Knopp, K. 73, 241, 245, 247, 267, 404, 448, 467, 477, 481, 483, 487, 509, 552. Kowalewski, G. 2. Kronecker, L., theorem of, 129, 485. — complement to theorem of, 150. Kummer, E. E. 241, 247, 260, 311. Kummer's transformation of series 247, 260. Lacroix, S. F. 555. Lagrange, J. L. 298. Laguerre, E. 555. Lambert, J. H. 448, 451. — series 448seq., 534seqq., 543. Landau, E. 444, 446, 452, 434, 502. Lasker, E. 401. Law of formation 13, 36. — of induction 6. — of monotony 6. Laws of arithmetic 5, 31. — of order 5, 28. Lebesgue, H. 168, 350, 353. Leclert 247. Left hand continuity 161. — differentiability 163. — limit 159. Legendre, A. BM. 875, 522. Leibniz, G-W. 1, 103, 131, 193, 457. —-— equation of, 914. — rule of, 131, 316. Length 169. Le Roy, E. 473. Lévy, P. 898. Limit 62, 462. — on the left, right, 159. -— upper, lower, 90—91. Limit of a function 159, 403—4. — of a sequence 62. — of a series 101. Limiting curve 330. Index. Limiting point of a sequence 87, 394. — greatest, least, 90. Limit theorems: see Abel, Toeplitz. Limitable 462. Limitation processes 464—476. — general form of, 474. Lipschitz, R. 368, 371. Littlewood, J. E. 407, 478, 501. Loewy, A. 2, 4, 10. Logarithms 55—7, 211 seq., 420. — calculation of, 23, 198, 254—17. Logarithmic differentiation 382. — scales 278seq. — series 211seq., 419seq. — tests 281—4. Cauchy, Machin, J. 253. Maclaurin, C. 523. Main criterion of convergence, first, for sequences, 78. — for series, 110. — second, for sequences, 82, 85, 393, 395. — for series, 126—7. — third, for series, 96. Malmstén, C. J. 316. Mangoldt, H. v. 2, 350. Mantissa 56. Markoff, A. 241, 242, 265. Markoff’s transformation of series 242 to 244, 265 seq. Mascheroni’s constant: see Euler's con- stant. Mean value theorem of the differen- tial calculus, first, 164. — of the integral calculus, first, 167. — second, 169. Measurable 169. Mercator, N. 104. Mertens, F. 321, 398. Method of bisection 37. DMittag-Leffler, G. 1. Mobius’ coefficients 446, 451. Modulus 7, 390. Molk, J. 564. Moments, Stieltjes’ problem of, 554. Monotone 15, 43, 162. — fully, 263 —4, 305. Monotony, p-fold, 263—4. — law of, 6. de Morgan, A. 281. Motions of « 160. Multiplication 6, 30, 48. — of infinite series 146seq., 320seq. — of power series 179. — term by term 68, 135. Napier, J. 56. Natural numbers: see Numbers. Index. Nest of intervals 19, 394. — of squares 394. Neumann, C. 15. Newton, I. 1, 104, 193, 211, 457, 564. Nielsen, N. 564. Non-absolutely convergent 136, 3896, 435. Novlund, N. E. 523. Null sequences 16, 43 seq., 58—61, 70, 72. Numbers: see also Bernoulli's numbers, Euler's numbers, — complex, 388seq. — irrational, 22 seq. — natural, 4. — prime, 13, 445seq., 451, 553. — rational, 3 seq. — real, 32 seq. Number axis 7. — concept 8. — corpus 6. Number system 8. — extension of, 10, 32. Numerical evaluations 77, 232-273, especially 247—260. Odd functions 173. Ohm, M. 184, 320. Oldenburg 211. Olivier, L. 124. Open 18. Ordered 5, 28. Ordered aggregate 5. Orstrand, C. E. van 187. Oscillating series 101, 102. Oscillation 159. Pairs of tests 308. Partial fractions, expansion of elemen- tary functions in, 205—38, 239, 877, 419, 528. Partial products 104, 224. Partial sums 99, 224. Partial summation, Abel’s 313, 397. Partitions, number of, 553. Passage to the limit term by term 338seq.: see also Addition, Sub- traction, Multiplication, Division, Differentiation, Integration. Period strip 413seq., 416, 418. Periodic functions 200, 413 seq. Permanence condition 463. Perron, O. 105, 475, 478, 551. z 200, 230. — evaluation of, 253—4. — series for, 214, 215. Poincaré, H. 522, 541, 548, 555. Poisson, S. D. 523. Portion of a series 127. Postulate of completeness 33. 569 Postulate of Eudoxus 10, 26, 33. Powers 47—48, 51seq., 423. Power series 151 seqq., 171 seqq., 401 seqq Prime numbers 18, 445seq., 451, 558. Primitive period 201. Principal criterion: see Main criterion. Principal value 420, 421, 423. Pringsheim, A. 2, 84, 94, 175, 221, 291, 298, 300, 301, 309, 320, 399, 491, 564. Problems A and B 76, 105, 230 seqq. Problem of moments 554. Products 30. — infinite, 104, 218—229. — with arbitrary terms 221 seq. — with complex terms 434seq. — with positive terms 218 seq. — with variable terms 380seq., 436 seq. Pythagoras 11. Quotient of power series 182. Raabe, J. L. 285. Rademacher, H. 318. Radian 57. Radius of convergence 151. Radix fractions 35 seq. — yecurring, 37. Ramanujan, S. 553. Range of action 463. — of summation 398. Rapidity of convergence: see Conver- gence. Ratio test 116—7, 227—8. Rational functions 189seq., 410seq. — numbers: see Numbers, Rational-valued nests 26. Real numbers: see Numbers. Rearrangement 45. — in extended sense 142. — of products 227. — of sequences 45, 68. — of series 186seq., 318seq., 898. — theorem, main, 143, 181. — application of 236—7. — Riemann’s, 318seq. Reciprocal 30. Regular functions 408. Reiff, R. 104, 138, 457 seq. Remainders, evaluation of, 250, 259, 527, 536—540. Representation of real numbers on a straight line 32. Representative point 32. Reversible functions 163, 184. Reversion theorem for power series 184, 405. Riemann, B. 166, 318, 319, 363. 570 Riemann’s rearrangement theorem 318 to 319. Riemann’s theorem on Fourier series 363. Riemanw's C-function 345, 445, 492, 533, 543, 561. Riesz, M. 444. Right hand continuity 161. -— differentiability 163. -— limit 159. Roots 48seq. —-- calculation of, 257—8. Root test 116—7. Runge, C. 564. Saalschiitz, L. 184. Sachse, A. 353. Scales, logarithmic 278 seq. Schevk, W. 239. Schlomilch, 0. 121, 287, 320. Schnee, W. 481. Schréter, H. 204. Schur, I. 267, 483, 552. Section 40. Seidel, Ph. L. v. 834. Semi-convergent 522, 541. Sequences 12. — bounded, 14. — complex, 333seq. — convergent, 62—76. — divergent, 63. — infinite 13. — null, 16, 43seq., 58—61, 70, 72. — of functions 327 seq., 429. — of points 14. — of portions 127. — tational, 19. — real, 42 seq. Series, alternating, 181, 250, 262 seq., 6 — asymptotic, 540 seqq. — binomial, 127, 208seq., 423 seq. — Dirichlet, 317, 441 seq. — divergent, 457 seq. — czponential, 148, 191, 411. — faculty, 446seq. — for trigonometrical functions 198seq., 414 seq. — Fourier, 350 seqq., 493 seqq. — geometric, 111, 179, 189, 472, 510. — harmonic, 79, 112, 115—6, 118, 150, 237, 238. — hypergeometric, 289. — infinite, 98seqq. — infinite sequence of, 142: see Di- vichlet series, faculty series, and Lambert series. ~-- Lambert, 448seq., 534 seqq., 543. — logarithmic, 211seq., 419 seq. — of analytic functions 428. Index. Series of arbitrary terms 126seq , 312seq. — of complex terms 388seqq. — of positive terms 110seqq.,274seqq. — of positive, monotone decreasing terms 120seq., 294 seq. — of variable terms 151seq., 326seq., 428 seq. — transformation of, 240seqq.,260seqq. — trigonometrical, 350 seq. Sierpiriski, W. 320. Similar systems of numbers 9. Sine 198seq, 384, 414 seq. Sine product 384. Squares, arrangement by, 88. Steinitz, E. 398. : Stizltjes, Th. J. 238, 802, 321, 541, 550, 554 seqq., 562. Stirling, J. 240, 448, 532. Stirling's formula 531seq., 543, 546. Stokes, G. G. 334. Stolz, 0. 4, 37, 74, 85, 311, 407, 564. Strips of conditional convergence 444. Sub-sequences 44, 90. Sub-series 116, 141. Subsidiary value 421. Subtraction 5, 29. — term by term 46, 68, 135. Sum 29. — of divisors 451. — of a series 101seq., 462. Sums of columns, of rows, 144. Summable 462. — absolutely, 515. — uniformly, 497. Summability, boundary of, 493. Summation by arithmetic means 478seqq. — of Dirichlet series 492 seq. — of Fourier series 494 seq. Summation, index of, 99. — range of, 398. Summation problem for asymptotic series 548, 553 seqq. Summation processes 464—476. — commutability of, 511. Sylvester, J. J. 180. Symbolic equations 183, 525, 528, 558 seq. Tangent 202seq., 417seq. Tauber, A. 486, 500. Tauberian theorems 486, 500. Taylor, B. 174. Taylov's series 174—5. Terms of a product 219. — of a series 99. Term by term passage to the limit: see Passage. Index. Tests of convergence: see Convergence tests. Theory of le marks on, 298—305 — systematization of, 305— 311. Toeplitz, 0. 72, 474, 491. Toeplitz limit theorem 72, 391. general re- Transformation of series 240 seqq., 260 seq. Trigonometrical functions 198—208, — calculation of, 258—09. Trigonometrical series 350 seq. Ultimate behaviour of a sequence 14, 45, 92, 103. Unconditionally convergent 140. Uniform continuity 162. — convergence of products 380. — of series 326seq., 428seq. — of Dirichlet series 442. — of faculty series 447. — of Fourier series 355. — of Lambert series 449. — of power series 333. — convergence, tests of 344seq., 381. — summability 497. 571 Uniformly bounded 337. Uniqueness of the system of real num- bers 32. Uniqueness, theorem of, 172. Unit 9. Unit circle 402. Value of a series 101, 460. Vicia, =. 213. Vivanti, G. 344. Voss, A. 322. Wallis, J. 18, 37, 219, 564. Wallis’ product 384, 531. Weierstrass, K. 1, 89, 334, 3845, 379, 394, 398, 408, 430. Weierstrass’ test for complex series 398 seq. — test of uniform convergence 345. — theorem on double series 430 seqq. Wivtinger, W. 523 seq. Zero 9. {-function, Riemann’s, 533, 543, 561. 345, 445, 492, RETURN TO: AEE | ATHEMATICS, STATISTICS LIBRARY Ba 100 Evans Hall 510-642-3381 LOAN PERIOD ONE MONTH 1 |2 3 4 5 6 All books may be recalled. Return to desk from which borrowed. 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