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TTTTTTTT tºrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrr. [- L L [−. [-. Eº E H E. T) >% | ? ON THE CHARACTERISTIC PROPERTIES OF SUM-FORMIUTAE IN THE THEORY OF - - - - - - DTWERGENT SERIES. A Thesis submitted to the Faculty of the Department of Titerature, Science, and the Arts of the University of Michigan for the degree of Doctor of Philosophy. by William Orville Mendenhall. I. TT. TTT. TV. W. WT. WTI. r T Tº N T E N - --- - - T. * rt - Q Tntroduction. Tefinitions of Sum. Consistency of Definitions. The Boundary –Value Condition. Fundamental Operations. Miscellaneous Remarks. Conclusion. BTBT.TOGRAPHY • E. Bore1 : / "Teçons sur les séries Divergentes", Paris, 19ol. "Fondaments de la Theorie des Séries Divergentes Sommables, " Tiouville 's Journa1, 5º seºrie (1896) | Vo1 . 2 º T. J. T" a Bromv7ich : "Theory of Tnfinite Series, " London, 19O8. #.. Buhl : 4 #. "Sur de Nouvelles Applications de la Theorie des Residues, " "Sur de Nouvelles Formules le Sommabilité . " Bulletin des Sciences Mathematiques t. 42 ( l9O7). E. Cesé(ro : "Sur la Multiplication du Séries, " Bulletin des Sciences Mathematiques t. 14 (1890). W. B. #'ord : "A Set of Criteria for the Summability of Diverg- ent Series, " Bulletin American Mathematical Soeiety Vol. 15 (1909). " On t, he Relation betWeen the Sum formula,5 of Hölder and Cesáro, " American Journal of Math- ematies, Vol. 32 (1910). G. # robenits : " Ueber die Teibnitzshe Reihe" Crelle " s Journal, Vol. 89 ( 1880) . G. H. Hardy: "Researches in the Theory of Divergent Series and Divergent Integrals." Quarterly Journal of Mathematics Vol. 35 (1904). "On the Differentiation and Integration of Divergent Series", Transactions Cambridge Philosophical Society, Vol. 19 (1904). A. Pringsheim: "Encyklopädie der Mathematischen Wissenschäften" T. A. 3. 1. I. Introduction. The theory of summability for divergent series, as developed within the past twenty years, has been direct- ed mainly along four distinct lines: (a) To assign a sum to a numerical oscillatory divergent series (/) 4, 7-2, -/- 232 + - - - of the type obtained by placing x= 1 in the power series (2/ 4, 7-4, x + 4, x^+ - - - it being assumed that the radius of convergence of (2) is unity. Thus the series / - / / / — / +- - - - obtained by placing x = 1 in the power series y - x - x* - x*z – - - has been the object of especial study." (b) To determine, as a function of x, an éxpression which shall coincide with the function f(x) defined by (2) within its circle of convergence, but which shall also serve to extend f(x) analytically in some region outside this circle. In this aspect, the study of divergent series belongs in reality to the general theory of analytic continuation, being * - ºvº- - - - - - - -- a------ - - --- * -------- distinguished merely in the special character of the method employed. ( 3 ). To study trigonometrio series (convergent or divergent) especially Fourier's Series, from the standpoint of general formulae encountered in connection with (a) above. (d) To determine, in case the radius of convergence of the series (2) is equal to zero, a function f(x) which, when considered in some region T containing the point x = 0 on its boundary, is developable formally for values of x in this region in the form (2), i. e. such that for values of x in T We have a, -º- fº, º – ºf 60,---, 4,-º,--- The determination of such a function f (x) is usually not unique. In the present paper we shall be concerned only with (a). In this connection many definitions of "sum" have been proposed and it will be our purpose to compare a number of the standard types in order to distinguish clearly those characteristics which are common to all. Also we sha 11 endeavor to discriminate between those àefinitions which, in the interest of a consistent and useful theory of divergent series, should be retained and those which should be relegated to inferior positions. The term "useful definition" as here employed we shall take with the following special, though arbitrary signifi- cance. Any definition of sum (case (a) ) is useful - when it assigns to the series (1) a sum, s, such that (5) s-º, Aa) where f(x) represents the function defined by (2) for and Where in passing to the limit, x is con- sidered to take only positive real value S. We vent; tıre to call such definitions useful in preference to others because We believe them best calculated to serve the general purposes of analysis. This property implies that if the series (1) be given, the series (2) which is to determine the function f (x) employed in (B) shall be the one obtained by supplying the successive powers of z into (l) in the particular manner indicated in (2). If such restriction be removed, - for example if with (1) we take instead of (2) the series (2/ (4. + 22, #. -- ** , )+(4, 7-4,” - - +4.2-,) X +- - - - Aſºº ſº pº-º/ *: . (2*) a, º ox + ox”--- +2, x *-2. X”--- **** -- it is evident that the corresponding f(x) becomes in general different from that determined by (2), and hence § as defined by (B) becomes altered. Logically the series (2*) or (2' ) may be taken as well as (2) for the indicated purposes: However, the functions f(x) which, according to varying methods of supplying the powers of x in (l), may - - - - - -- - - - - --- -- - - - - -- - - - - - - - - --- - - * Pringsheim: loc. cit, T, A, 3, 40, 4. be thus constructed and employed in (B), seem relative- ly unimportant. Emphasis must certainly fall upon some special type of power series (2) if we are to assign to (1) a unique sum s, consistent with (B), and it is but natural that the type preferred shall be (2). It seems clear, moreover, that a theory of summability thus con- structed will best serve the general purposes of analysis. TI. Definitions of Sum. Th the discussion of the various definitions the following notation will be employed. We shall represent the given series (convergent or divergent) by (3) tº # *, + 42 + - - - and let (4°) 3, - 42 +*, * - - - **** If (3) is convergent we shall represent its sum by S, i.e. Tf (3) is divergent s shall represent the value assigned to it; as Sum. - The definitions for s to which we shall confine our attention are as follows: - (*) (I) 3-ºxº #. (cessºro" in Which - 3” – S. + r. 8,..., +- "gº Sº-2 -y- (J) . of which as a special case, we have, When r= 1, rtrº-J --crº-1) n . So r) ( r+1 / (r-1-2) - - - (n+ n) D. - zz/ * Bulletin des Sciences Mathematiques (2d serie) tº 14 (1890) p. 119. (IT) s-ºr- ***** - + 3. (ITT) 3 = 4:4. gº - (Hölder”). in which S.." = #Fls, 3,4---- +S.) sº-J., sº sº.....sº) (TV) S-422. e T's, a y (Borel"). in which s(a) is defined by the following series (assumed convergent for all values of a \, St-J = Sez S, ºr *- *** *--- ~ – aſ º (V). s-ſ -č 4 (a)* (soreif. in which ua) is defined by the following series (assumed convergent for all values of 2: ) *(*)=4< **, *, +4, 24*, --- 3. 22,’ º - aſ (VT) s= ſe’, (a/~~ in Which **(*/ =(w, rº,+ -- ***..)+(** * * : *%-, Jºº,” ºr 9., Jºr-- arºo - ºr - ++ (TTT) s = ſ. & U (a/a (Te Roy) © º # in which V (*) = 3 ºr . A ** an extension of (W). * Math. Annalen, Vol. 20 (1882) pp. 535-549. f E. Borel; Lecons sur les Series Divergentes, Paris, 1901, - P. 97. f Toc. cit. pp. 98, 130. ++ Annales de la Faculte des Sciences de Toulouse, (2nd série) º t. 2 (1902) p. 217. 6. - º ſ (2: (WTTT) s= ſº X tº rºl, * = 2 /7/* +// * (I.e Roy ) or, in another form, - -> -4- –4. - - Z - - ? S = - Aſ ce ºf 2 - * zz/a/2.2. = / Zº º Besides the above mentioned formulas, Euler's transformation for obtaining from a given series one more rapidly convergent may, when applied to divergent series, give a convergent, one, and hence has been used to define a sum for such series. III. Consistency of above Definitions. Any definition of sum for a divergent series must be such that in the case of a convergent series it gives s = S, i. e. it must be consistent with the established definition of sum in such cases. This property of a definition is termed its consistency. We shall now establish the consistency of the above definitions by a uniform method based upon the following lemma: Lemma, Tet sa, sº , ss. . . . . . . . , 8... . . . . . . . . be a sequence - of quantities (real or complex) such that and let a!", aſ”, a!", O - .......aº, be a sequence of positive quantities dependent upon a parameter p (in- dependent of n). Also let it be supposed that; the - - eº ſº) Sºre SS.1011 3. _ 3 & 8- % -> ſº º' º - - - - - - -- - - º Bromºioh , Infinite Serie º -i. Tondon (1908) p. 55. Bromwich: log. Cit. has a meaning for every positive value of p in a given sequence P whose elements increase indefinitely to pos- itive infinity. If p be allowed to increase indefinite- provided that (A) Where - Where than 2. º .# 4, ź. Zay º,” - -- - - m is any fixed positive integer independent of p and (2) represents the largest integer equal to or less * = o ſ zº X' 4 Proof: S, –4, ** Where º *... = 2 - ſº/ -ºr M. º/ (?/, /a/ ºe AAJ D ... *s. -/ 34%, 3.” %, ’2%, .* - *// = -**- - -: 22-0 ºr. º ºw- ºgº * = * = ** , # lºl / /// 2, | > 4. > 4. | * = D. * =Z) - jºsº.3.4". - &- 7: so DA/ * f s = | & , , o, . 2,--~~. /*/ 2, 4. 5 ſº X, a " _ _ºe “” / – 21:2–1 T /*/ (3+ |&l) 7- £ - /// (7) ºg ** = 0 º, * = 2 , Z = **, *, +,---- Whence upon applying (A) of the hypothesis we have ºf 4-2 ºz. … º. 3.- & We begin by showing the consistency of (II). (7) Let 4'-' cº - Ez, "ea º a 2 and let P be the sequence 6, 1, 2, 3, . . . . . . Then 3. = Ser 3, # - - - * $4. A z- a º where so, s have the meaning assigned in (4). 1' ' ' ' ' ' ' ' • *p (A) is satisfied since - (*/- 0 and it; follows that ºr serºtº- + r * ~ *... s. -z-z - cº-º – cºº A , , Whenever the latter limit exists. To show the consistency of (T) let P again be the (?) sequence 0, 1, 2, 3, . . . . . . and let Z, - / When n >A, *= *I *# t—tº when wº, aſ-, when n = A, - - º) (*J - then S.- Sº where & J 4. are defined as in (5) * Z." (A) is satisfied since *(*-ij- (ºrf– 1 J ****) - - -(º-A-2J , fºr-rº) ---ſº-ſº-w-J * Z/ + #2-# *--- ****** Ż-e- iſ rºl trº-2J - - - r + PJ A/ - // . *— + * * 7- - - - *-0---4---0) =0 zºº, "riº (**) (prº-1) ºrſº-y- -(t+/- º Thus We have 2 : - S.” 2 . 2:-} S. = ** #. = ºr S. Whenever the latter limit exists, i. e. when (3) is convergent. To show the consistency of (IV) let P consist of all the values in the continuous domain A = 0 ºn 6 16t; (A) 27 - Ž 2 - - - - – 4, =#. Then S. - So f- *** * - = 4 3/2) /*A*** - - - - and (A) is satisfied since --- Zºº / / / / #7 7- - - - * : 7 (6) Z--> _** = 0. * ºr e^2 = 4a e sº-º: s. when (3) is convergent. It may be noted here that by the aid of the above lemma a sufficient condition may be obtained at once for the consistency of the following general definition * s = ºr 24's. 2. The condition is that; /*ſ - 4", 2% -- - - r * - O . , AP/ - L. %24- ºr a;”. – -- ºf */ To show the consistency of (W) we note that when (3) is convergent º * = 0. Whence applying the lemma, * 2: 2. if we place S.–4, 4 becomes zero, and if we place 4, Ž I (A) is satisfied because of (6). Thus We have (7) º: 27.2/= ± e 22(a) = 4 * This general form of definition of which (TV) is the special case corresponding to 24"– A * , has 27 ºt; been studied by Borel (loc. cit. pp. 92-95) and others, the quantities zz." being termed Weights. The defin- itions I, II, III also may be regarded as special cases of this form, 10. Moreover it appeared above that when (3) is convergent - *ſ - - - ºf * 2 "stay = S. Now if 2%. 2 Tºe (a/** 2: = <> - - ºf and if º a 3/a/ exists, we may show that /ºza, ** = -4--, * sº º and therefore, in view of the consistency of (TV), that (W) is consistent. Since S=4: 2 sta, and since 2 ”sſay - z, when a = o , we have S-24, - ſ' 4, 14. 34-1/2 But; # Rºsſal/– 2 /3 (al- s(20) Where 3, Ø j = 8, s, a *s, *, +--- - a * *- - - - Now 3, a J – S(a) = 4, 4- 42 a + 42 #7 ºf and if we define tº, (a J as the second member of the last equation. We have (8) 5-a, - /Te "…a, a Integrating by parts, we may write S-a. = |er"/"a, (a) /a/ - ſº-º/J. "...(wº/* = /er"/aeº-º/* /2 "/…/-...} Zx * This condition was omitted in Borel's discussion (loc. cit. p. 98) but was shown necessary by Hardy (Quarterly Journal of Mathematics, Vol. 35 (1904) pp. 29, 30). - 11. Introducing (7), we now have 8–4 = J. 27"/…/ –42/*. Since lºo- ſº a We have 3– 47%alaa. Q. F. D. The definition (WT) is also consistent, for, when the series (3) is convergent to the value S, so also is the series (ue #~~, …, , )*(*****, + -- +44-, J-H42, 4- - - - -26. L)+- - º and by (W) we have S - J. º "...ejº Likewise (VII) may be shown to be consistent by reducing it to the case of (V), for it is simply the application of (W) to the series zze + o f - - - 0 + 4, 4-0 + - - + 0 + 42 f 0 7" " " Where 7% –/ zeros are inserted between each term and the succeeding one in (3), and if (3) is convergent this series is convergent to the same sum S. The consistency of (VIII) will, appear as soon as ee> / - º- cºo it is shown that Zºe 4/ ºf € º, af waja’a = / & 7.2/. z-- / © whenever the latter limit exists, for When (3) is con- º 4. vergent / *Tºtaja'a = 3. No.7 *…, –4– º, * 2°:// zº-e Z * /, - * x ** 2: may be written -º- Z aſ Tº */a/a4 Furthermore º _*, *** o 2ſ – aſ º _4… / 2. ‘e’ way 42 = 2^2 / 2. ‘e _* 22 (a /24 %. Bromwich: Infinite Series, p. 300; Hardy: Quar. Jour. of Math. Vol. 35, p. 36. 12. and * Abel's test for uniform convergence is applicable ee - 2, - to this last integral since / *T* (a) A is convergent and since when x is positive and a > / , the function *** 2x * 2--- decreases as ox increases, and is equal to / When 2 - / • Therefore tº ſº ºf - * --> - Jºe cº - - - º ſ 2 * * ºf wa, 42 = / *:: *** º *2 *…* = / e - "24/a/2/a. - z - nº-ºº: Z - Moreover % - / 2 * 2° * 2° 24′2.2/-, – Ze ** 2: Jº'a A = 0 because the range of integration is finite, with which the proof is complete. TV. The Boundary Value Condition. The definitions in Section II have been seen to be similar in the one respect that they all are consistent. We shall inquire now which of them satisfy the boundary condition mentioned in Section I, viz.: * This test in the form here used is as follows: "The º - integral 4 f(x,z) Ax, ax is uniformly convergent in an º - interval (2,30 , provided that A 24x/4'x converges and that, for every fixed value of y in the interval (2/3), the function 7°(x, y) is positive and steadily increases 3.5 x increases, while f (a, y) is less than a constant K. (independent of y)." (IB) 3-4: f(x,y A z real and positive. Where (9) fix) = 4, 4, x - 4, x - - - - (convergent by hypothesis for l x 1 - / ). First we shall show that the definition (I) satis- - º o fies (B), i. e. 4. XX 4, x'= ºg When the X = / : º -77 - --> ADº" º, latter limit exists. This may be done by the aid of our Sº") º tº º lemma, as follows: Tet; be the S, of the lemma, then ~~~ S. exists by hypothesis. *22 – acº - Place A = 7-4 so that as x ranges in value from z z. z (6-4-2) p ranges from (* J , ;4: Z z-ee. Also place ZO. ( – 4/-a.” --> M. = ~Uri * g.- : *** - sº A T º {r} 1 – 4)" - Tºsz tº 2. P. ( A X D. X * = 0 —trº) T. *A* (Z-29 º * Frobenius (crelle's Journal, Vol. 89, p. 262) proved this for the special case r- 12 i. e. º X w.x." ~ a2,4-(3,-sº -- + š, ) eº 21 + *72 -42 2-º- ºr which is the case of definition (IT). For a proof of the general case different from the above, see Bromwich, Infinite Series, Art S. 51, 123, 152. 14. and 4… & = ~ 2. 4, X* Zº-e-> Condition (A) is satisfied for * Dº (, ; " aw” _/ 1*- Zºº'ſ -à, *** - J - . [. J J * 1) (a - - - I ºn º -Z, ºr ºu-4/rº-3'---- ****{-4) A=~ / , , º, 7. - (rºr'] [r raw - - r + ſ^J/ ſº - * **ſ-4) - - - - 1%// (1-4)] Which is evidently zero since the denominator becomes infinite with p. Whence *… 3. = ~. 3. - 2–~~ * - --> i. e. __, 2 a.k"— 2. - º ...” X -/ * = - 2 ºr 22 We turn next to definition (V) and shall show that * 2 . , 22 ^^ - (10) º / A "* (a x)^a = /? 4.4/4'a, thus proving */ - that (W) satisfies (B). Let ~£a = 4, then iſ rºw,” – 4- Zºzº 47/4 Again place # = / / 4 and * tºgral becomes (//4// Te”uy,” 2. //**// 2-4.2-7. */4 To establish (10) it suffices to show that its first member converges uniformly for values of x in the interval (7 & 2. ~ x ~ / , or that the integral last Written converges uniformly for values of 4 in the interval from - - - - - ----- --- - --------- - - - - - - - - - ----- + Bromwich, loc. cit. Art. 111. 15. =a_2 - ? oza 4 – /. Now the integral 4. -č’ */4 is convergent by hypothesis, 2%is positive and steadily decreases as y increases, and is equal to Z for any fix- ed value of 4 when y = a * Therefore Abel's test; applies and the */ 2”// is uniformly convergent in the interval in question with which the proof becomes complete. To see that (TV) satisfies (B) we note that - cº 3 = 4. * "3 (a) = a, , A 47 'º, Zaya'a (11) Cſ- cºo as appeared in the derivation of (8). In the series (9) let, 20, S., - we – w, x + 42 x /- - - - *** * 2, 21 and let s, a 1 = 8.2 ° S., a *.s., 4, 7 - - - (corresponding respectively to 8, and 5 (a) relating to the series 2, #-4, 4- 4 +---). Now 4 (a) being convergent for all values of ox, the same is true of w (a x) for all values of X and of . Furthermore if aſa) is convergent 8(a) is also. This appears from a consideration of the function */a/- 2 "Z / 2. "... /a) (a +4.7 © which is analytic throughout the plane and whose develop- ment in power series is therefore convergent. To obtain this development. We have <2% (a) = 4, a y + 4*/ - M -// (*- *A*) = a "º)+ A*/ º 16. % - ſo = S. = S Whence 22 (0) = 42 = 82, ~z; † / 4, to y +% / *, * 52 * * * * - - - - * * - - - - - - !”). -f - - - <2**6) = 20° */o/* A" %) = 14, 7 S., -, - 5. * <2*/a) = s. As a , 3, 3} + - - - = S(a) ---- - - - Therefore S(a J is convergent. This proposition being true for all values of 21, we may say that when 22 (a x/ is convergent $x (*) is also convergent. Then placing - * -é ". S. = 2: = <> se (a/, Tre haye. ºc — a rºle S. - 42 = ſº "S. (a) , = / 4. /3. (a) – 5.4°//a = Z 2 "Zoº 4 x*, *, 3, 2-7 = x/ "... (axy 2. - and º: (5. - 4./ = 4; X /*-42/72. But by (10) -º- ſº 2.2 x/2/4 = A 7, ſay.” Therefore -º-, (3, -4, J = Z "a - 2/a/~ which integral is equal to 3–44, from (Ll) and thus we see -º-, 4-, erºs./a) = 4… e. "stay X = / ºf = <> ºf ~ -º which is the requirement of (B). The definition (VI) is not obtained by multiplying the successive terms of the series to be summed by the successive terms of the expansion of 2 ”, lout; an arºit; r- ary grouping of terms is made before the terms of the 17. exponential series are introduced. This operation results in obtaining a sum which in general is not unique, but which depends upon the value assigned to K. This appears from an example. In case of the series / – 7 / / – / /- - - - if k = 2 (or 2n where n is a positive integer) the series becomes after grouping 0 7- 2 A- Z2 y— — — — and 3 =0, but if K = / ſº.2n+1) the series remains the same as before and u, (a J = < (a ), whence S has the same value as that given by (W), namely # * Since the sum is not unique (B) is not satisfied in this case, and hence not in general by (VT). The question remains open whether or not definitions (VII) and (VIII) satisfy (B). W. Fundamental Operations. Besides being consistent and satisfying the bound- ary value condition it is desirable that the sum assigned to a divergent series be such that the usual laws of trans- formation of convergent series will be preserved. The laws of this type which we shall consider are as follows: _ (C). If s represents the sum of the divergent series X u, , 242, 3" iſ a , n-º-º-º-º: ** ºf zº) A - * = pſ should exist by the same definition and be equal to and conversely. 3-(4, 7-4, 7-4, 7 - - - - +,-, ), y 18. (D). If for a given definition ºf sun, sh repres- ent the sum of the divergent series 2 *- and Ps the sum of another divergent series ºf º, then 8, #: S. --> ra should represent the sum of 4. (*, + M. J. (E). If sº and sº have the meanings just given, then the product sº se should (at least after certain further restrictive conditions upon td., M. . analogous to those imposed when two convergº series are multiplied together) represent the sum of 2. 10, where * = u, º, ø u, V., P --- * u, , Ø rººk We begin by considering condition (C). It is not apparent that this condition is satisfied by any of the definitions of Section II. In order to be assured of (C) it is usually necessary to make an additional hypothesis upon beyond those imposed in the original statement of the definitions. Thus, considering first definition (TV), we shall now show that if, in addition to the assumed existence of 4… 3aºhe series (3) is such that - … "sº-º-º-º: sº ſº. then (c) is satisfied. Thfact if we place 3. = 24, 7- *, +- - - - - + 4* +, * - º - / 4. Af 2- (12) 3/3/ = S + 3,' a 4– S. 37 ----- We then have . --> • - Zsóay 3 = - ºf 2/ º - º 5 =/2 *(*) 42 exists and is equal to Sºu. The existence of (C) under hypothesis (b) has been shown by * In the converse proposition the statement of (a) is however unnecessary since the convergence of the º integral ſº, (a/4% is a sufficient condition that O - - ºf 2ſ – ae 2 (*) = 0. (See Bromwich, Tnfinite Series, p. 272; Hardy, Quarterly Journal of Mathematics Vol. 35, p. 30. ) -- 21. Hardy.” It seems probable that definition (I), and hence (II) satisfies (C), though it does not appear to have been proved. It is obvious that (D) is true for all cases, from the principle that the sum of two limits is equal to the limit of the sum. In connection with (E), we have, for definition (T), Cesáro's theoremſ ſet r be smallest integer for which 4… 3. exists ( S" and Z" defined as above), then r is cººled the degree of indeterminacy of the series Whose terms occur in sº . Cesaro's theorem is that if X.4, and X w. are two series whose degrees of indeterminacy are r and 5 respectively, then the product of the two series has a degree of indeterminacy at most equal to rº-sº-ſ. It appears that none of the ot; her definitions have been shown to satisfy (E) without further assumptions %. Quarterly Journal of Mathematics Vol. 35, pp. 30–32. Here Hardy calls attention to the fact that "the notion of absolute summability on which M. Borel lays consider- able stress, does not, here at any rate, give us any real assistance. " + Bulletin des Sciences Mathematiques, 2nd séries) t. 14, (1890), pp. 114-120. 22. as regards ll, and V. Borel"in considering (V) in this connection has defined an absolutely summable series as *_a i º) one for which ſº |a (*)]/3 is convergent, and has shown that if two absolutely summable series be multiplied to- gether by the usual process the resulting series will be absolutely summable, and its sum will be the product of the sums of the original series. Hºras' has added that if one only of the original series is absolutely summable the same result follows excepting that the sum- mability of the product is not necessarily absolute; also that if both of the original series are summable, , not absolutely, and if the product series is summable, its sum is equal to the product of the original sums. WT. Miscellaneous Remarks. It has been shown that definition (III) is coexten- sive with (I) both as to the value assigned by it to s, and as to the range of its applicability to a given series tº For that reason it was not considered in the above discussions. The lemma stated in Section IIT will serve in show- ing a relation between the results of Euler's transform- - * - -- - -- - - --- - -- - - - - - - -- - - - - - -- - - -- * Leçons sur les Series Divergentes, pp. 104-107. f To c. cit. º) • 43-45. f W. B. Ford, American Journal of Mathematics, Vol. 32 (1910) pp. 315-326. W. Schnee, Mathematisch Annalon, Vol. 67, (1909) pp. - 110–125. 2 3 º ation and (V) . In order to show this relation we first note that, by definition, Euler's transformation. When performed on the series (2) gives (E) , , La. -(Fay 7 - (< */7” --- / / #-X in which = ~ * f / + x and º * - - - - - - - - - - - - -- - The relation. Which we desire to show is as follows: Having given a series of the form (2) whose radius of con- vergence is unity, if it can be shown that the correspond- ing series (E) is convergent when x + 1, and that its terms 4, - - - may be rearranged at When expressed in terms of 4, 2, 4, */ pleasure then (2) is summable by Borel's definition (W), and Borel's sum is equal to Euler's sum. (E) being convergent, denote its sum by S. ... We cº *. cº - ºg - are to show that the integral ſº *::: aſ a exists * = 2 o and is equal to &. For simplicity we write Fe, Æ, E2, - - - for 4, Æa, £2,--- . Then the series (E) becomes when E. A. *~... --- Applying the lemma, let us place 3... = ºz, Then £= O since: # =o ((E) being convergent). Also let (?) #":2* and replace p by 2: . Then 4, ºn 7 - 24. condition (A) is satisfied for (2a,” - º- 22a’ 4. raz, ºz º. * - -o - --- =A7 2a: Therefore A. … • 2 * / A-. Aſ a , Æ, a” –4 ºr 2 ~4– *-2 - - - - O º /* * † - 2, 2% 74- y Using the lemma again, let - E. Æ. A. = ~2 -/- 4 --- ** – . S. 2 74- º: 7- +/- _2” +, Ther. A- Sg . Tet; */ have the same value as before and again replace p by ox. Condition (A) is satisfied and We have (*) 4- 2 ºf "sºza)= &= º- in which 3(.2 a y = Se f S, 2-2 ºr s.º +- - - - Now if we write 4x = A. (13) takes the form 2 */ - 2– - º (2.a'ſ _2 , 2-4-2.22) = 0 in which 42 a = *, ******* *** * *- aſ ºt adº - - Hence applying the results in Section V (viz.: if - º - ** , , , , – - - - ºf - - ºr - ºf e *(*/ = ? a-ºº: e. 3/21/2 S. * / 4. 214,4'- 5) and noting (14), we have (15) ſerºſa 2,42-) =/º/A.A.- . A # ---/*-5 - 2 º 40 A. *37 7- - Aſ Expressing AE2, A, , 22, -- in terms of 22, 2, , 4-, - - and rearranging terms (which we may do by hypothesis) We have * * * * * *g, * --- =^{4, re. ~ **-gz-- and therefore substituting in (15) we obtain * — a cº-e - 22 ſe 2. *::: ** = 32. 2-2. Which is the desired result. The cases of especial interest are those in which the series (E) converges owing to the vanishing of all the coefficients, P. after a certain point, in which cases the conditions of the theorem are fulfilled. This situation arises in particular when the series to be summed is of the alternating type tº e - 4, # 42 – 2: 2 + - - - 2 * * /*a*, and the successive differences among the quantities 4, w, 4-, --- eventually become zero. As an illustration consider the series / - 2 + 3 − 4 + 3 - - - // 2, 3, * 3: - - - –/, –Z –Z ~. - - a O 2, 0, - - - AE, A, - A 2- - z' / / º 4–2- -A- ~~~ +/- - - - - - - - - = 2 - 2- 44- * 2. 24- 2,4- In such cases this method is probably the shortest one for finding s. VII. Conclusion. We have found that the definitions of sum for divergent series proposed by Cesáro, Hölder, Borel and Le Roy all are consistent, that those of Cesáro, Hölder 26. and Borel (I-VT) satisfy the boundary-value condition: 3 = %, f(x), While the question as to whether those of Te Roy (VII, VIT) satisfy the latter condition is still open. Of these eight definitions that of Borel (7) is the only one which, besides having the two characteristics just indicated, has been shown to preserve relations (C), (D), (E) of Section 7 after certain further simple hypotheses are made. However Cesàro's definition (I) has been seen to satisfy them all except (q) and it is possible that it may satisfy (C) with a further hypothesis no more restrictive than that added in the case of (V). º - º - ºil - - - ºffl i. - - - - - - - - - - - - - - -º-º- --~~ - --------- --- --- - ------- - ------------- ------- -