CORNELL UNIVERSITY LIBRARIES Mathematjct Library Whfte Ha»f CORNELL UWVERSnY LIBRABV |_3 1924 059 413 058 DATE DUE A 00 ft 7noA APR 4 iUU4 CAVLORO M Cornell University Library The original of this book is in the Cornell University Library. There are no known copyright restrictions in the United States on the use of the text. http://www.archive.org/details/cu31924059413058 Production Note Cornell University Library pro- duced this volume to replace the irreparably deteriorated original. It was scanned using Xerox soft- ware and equipment at 600 dots per inch resolution and com- pressed prior to storage using CCITT Group 4 compression. The digital data were used to create Cornell's replacement volume on paper that meets the ANSI Stand- ard Z39. 48-1984. The production of this volume was supported in part by the Commission on Pres- ervation and Access and the Xerox Corporation. Digital file copy- right by Cornell University Library 1991. THE UNIVERSITY OF MISSOURI STUDIES MATHEMATICS SERIES VOLUME I NUMBER 1 ON THE DEFINITION OF THE SUM OF A DIVERGENT SERIES BY LOUIS LAZARUS SILVERMAN, Ph.D. Instructor in Mathematics in Cornell University ^ ^, Formerly Instructor in Mathematics in the University of Missouri -it ' t^V ,P^^'i^' UNIVERSITY OF MISSOURI COLUMBIA, MISSOURI April, 1913 ON THE DEFINITION OF THE SUM OF A DIVERGENT SERIES THE UNIVERSITY OF MISSOURI STUDIES MATHEMATICS SERIES VOLUME I NUMBER 1 ON THE DEFINITIONT OF THE SUM OF A DIVERGENT SERIES BY LOUIS LAZARUS SILVERMAN, Ph.D. Instructor in Mathematics in Cornell University Formerly Instructor in Mathematics in the University of Missouri UNIVERSITY OF MISSOURI COLUMBIA, MISSOURI April, 1913 Press or The New era printing coyMNi Lancaster, pa. CONTENTS Page § I. Introduction i § 2. Historical Resume 3 § 3. averageable sequences i5 § 4. Product Definitions 23 § 5. On Certain Possible Definitions of Summability 33 § 6. Definitions of Evaluability 46 § 7. Applications 63 § 8. Tests for Cesaro-summability 76 § 9. Theorems on Limits 83 § 10. Conclusion 89 §^I . INTRODUCTION * The series «o + Mi + M2 + • ■ ■ is defined to be convergent whenever L (mo + Mi + • • • + m„) exists; and the value of this limit is called the sum of the series. If this limit does not exist, the series is said to be divergent. Some writers call a series divergent only when L (,Uo+Ui-\ -\-Un) = 00 ; all series which neither converge to a finite limit nor diverge to infinity are then called oscillatory.! The present considerations are limited to series which are oscillatory. We shall follow, however, the terminology of most writers| by calling divergent all series which do not converge; stating expressly, if necessary, when a series diverges to infinity. A necessary condition for the convergence of a series is L m„ = o. Thus only a limited number of series can be dealt with. It is accordingly desirable to extend the definition of the sum of a series, so as to include a larger number of series with which we may deal rigorously. Our object will be to retain the class of convergent series, and to add to that set, by means of a more general definition, as large a class as possible of series which are not convergent. In order to be able to deal with these new series, however, we shall wish to preserve several funda- mental properties of convergent series. We shall, in fact, demand the following fundamental requirements of any general- ized definition of the sum of a series: * This paper was accepted as a dissertation by the Graduate Faculty of the University of Missouri in May, 1910, in partial fulfillment of the require- ments for the degree of Doctor of Philosophy. t Bromwich: An Introduction to the Theory 0} Infinite Series, p. 2. t See e. g., Goursat-Hedrick: Mathematical Analysis, p. 327. I 2 INTRODUCTION (i) The generalized sum must exist, whenever the series converges, (ii) The generalized sum must be equal to the ordinary sum, whenever the series converges, (iii) Each of the series I Mo + «1 + M2 + • • • 1 U1 + U2+ ••• has a generalized sum, whenever the other has, and t = s — Uojii s and / are their respective sums, (iv) If each of the series I Mo + Ml + «2 + • • • 1 »0 + f 1 + »2 + • • • has a generalized sum, A and B respectively, then the series (mo + fo) + (mi + f 1) + {Ui + Vi) + •• • has a generalized sum which is ^4 + B. (v) If the series Mo + mi + Mj + ■ • • has s for its generalized sum, then kuo + kui + ■ • • has a generalized sum which is ks. I wish to express my gratitude to Professor E. R. Hedrick for his interest in my work, and to acknowledge my indebted- ness to him for many helpful and important suggestions. I am also indebted to Drs. W. A. Hurwitz and H. M. Shefler for many suggestions and criticisms. § 2. HISTORICAL RESUME * The earliest interest in divergent series centers about the series I - I + I - I + ■••. If we assume that this series has a generalized sum s, then the series, obtained by dropping the first term, — i + i — i + i--- must, by the third fundamental requirement of page 2, also have a generalized sum which is obviously — 5. We have then, s — 1 = — 5or5 = |. Thus, if the series is to have any value at all, that value must be |. And this is precisely the value which Leibnizt was led to attach to the series, by different con- siderations. The sum of n terms of the series is o or i according as n is even or odd; and since this sum is just as often equal to i as it is to o, its probable value is the arithmetic mean, 5. This same value was later attached to the series by Euler,t in a more satisfactory, though not entirely rigorous manner. " Let us say that the sum of any infinite series is the finite expression, by the expansion of which the series is generated. In this sense, the sum of the infinite series i — x + x^ — x? • • • will be 1/(1 -far), because the series arises from the expansion of the fraction, whatever number is put in place of a:."§ In par- ticular, i = I - i-Hl -1 + •••. *The best historical sketches are to be found in Borel: Leqons sur les Series Divergentes: Introduction, and in an article by Pringsheim given im- mediately below. t See Pringsheim: Encyclopddie der Math. Wiss., I, i, p. 107, note. t Instil. Calc. Diff. (i755). Pans, II (p. 289). SThis quotation is taken from Bromwich, loc. cit., p. 266. 3 4 UNIVERSITY OF MISSOURI STUDIES It is true, as has already been intimated, that none of the methods given above, to prove that the series should have the value I, is satisfactory from a theoretical point of view. But objections have been raised* to the result for practical reasons also. Thus, the series i— i + i — ! + ••• may be obtained from the expansion I + X + X^ I — x? and setting x = i, 1 = 1-1+1-1+ •••. To meet this difficulty, Lagrangef observed that we should write 1 -\~ X ; ; ■„ =l+0-X — OC^ + X^ + O-X* — X^+---, I + X + X^ SO that for a; = i , we have f = I+0-I + I+0-I + ---. If we now follow the method of Leibniz, we see that the sequence corresponding to this series has, out of every three succeeding terms, once the value o and twice the value i ; its sum is accord- ingly f • Thus, Lagrange has removed the practical objection. Moreover the above method has been put on a rigorous theoretical foundation, by means of the following proposition, t which is a generalization of Abel's theorem : Theorem a:§ If Sn = uo + Ui + u^ + ■ ■ ■ + «„ and J V so+ Sl+ •■• + 5n "j _ * By Callct. See reference immediately below. t Rapport sur le Memoire de Callet, in: Memoires de la classes des Sciences maihemaUques et physiques de I'lnstitul, t. III. t Frobenius: Journal de Crelle, t. 89, p. 262. § Theorems embodying new results we shall indicate by numerals; all other theorems will be lettered A, B, C, • ■ ■. DEFINITION OF SUM OF A DIVERGENT SERIES 5 then n L S U^X" = S. x=l Thus, in the case of the series i — i + i — i + --, T ^° "^ ■^i + • • • + ^" _ 1 n=» n 2 ' and accordingly L (j _ ^ + ac' • • ■) = i; so that we may define the value of the series i - i + i • • • to be L (i - a: + x'' — x^ + • ■ •). or what amounts to the same thing, J ^0 + ^1 + • • • + ^n n=co W whenever the limit exists. The first mathematician actually to carry through the de- finition was Cesiro,* who approached the subject from another standpoint. Cauchy has defined as the product f of two series I Wo + «i + ••• \ Vo +vi+ ■•■ the series UoVo + (utiVi + UiVo) + {uqV2 + MiVi + U2V0) + • • • ; this definition being justified by the theorem, due also to Cauchy, that the product series thus defined of two absolutely convergent series, is itself absolutely convergent. MertensJ has generalized this theorem by proving that the Cauchy product of an abso- lutely convergent series by a simply convergent series is con- vergent. The product of two simply convergent series may, however, be divergent. Cesiro has studied the divergent series which result from the product of two simply convergent series, and has obtained the following remarkable theorem : * Bulletin des Sciences mathemaliques, t. XIV, 1890. t We shall later refer to this as the Cauchy-product. t Journal de Crelle, t. 79, p. 182. 6 UNIVERSITY OF MISSOURI STUDIES Theorem b : Let the two series {Mo + Ml + M2 + • • • fO + l*! + »2 + • • • converge to u and v respectively, and let \Wn = (Wol'n + MiJ;„_i + ■ • • + M„Do) I Sn = Wo + Wi + • ■ ■ + Wn then T ^0 + 5l + • • • + 5„ Li 1 = u • V. B=« W + I The two theorems which we have stated justify us in stating the following definition: Definition:* // 5„ = Mo + Mi + M2 H + m„, the series mo + Mi + ••■ +M„+ ••• is summahle and has the value s whenever T ^0 + ^1 + • • • + in lu r— = s. n + I Let us now proceed to show that this definition satisfies the fundamental requirements of page 2. To this end, we shall prove the following theorems. Theorem c:t If a series converges, it is summable, and the two definitions give the same sum. Let Sn = Wo + Ml + • • • + w„, and L 5„ = 5; we shall prove that T 5o + ii + • • • + 5„ 1j j = 5. We have: So + Si+ ■■■ ■\- s„ s W + I (5o-5) + (5i-5) + --+(5,-5) + (5,+i-5)H |-(jn-5) W + I ^ |5o-5| + |5i-5|H h|5<,-i-5| Ua-^l-j h|5n-.r | ~ n + I « + I * Cesdro calls series of this type simply indeterminate. t By this theorem requirements (i) and (ii) are satisfied. DEFINITION OF SUM OF A DIVERGENT SERIES ^ Since Li 5„ = 5, we can take g so great that I 5< — 5 | < e / 2, i > g. Having chosen this g, let L be the largest of the numbers, 5< — 5 I, i = o, I, 2, • • ■ g — I, Then we obtain: ^0 + ^1+- • • + Jn _ ^ qL {n — q-\- i)e gL e w + i ^~n-\-i^ 2(w + i) "^w + 1 + 2 We can now choose « so large, « > r, that < M + 1 2 and hence, ^0 + 5l + • • • + J„ I — s\ < e, n > r. n + I Theorem d:* Each of the series I «0 + Ml + Mz + • • ■ Ml + M2 + • • • is summable when the other is; and s and t, their respective sums, are connected by the relation s — uo = t. We shall prove only one part of this theorem, the method for the second part being exactly the same. We begin by proving the following fact. Lemma: If the sequence so, Si, • • • s„, • • • is summable and has s for its sum, then the sequence S\, 52, • • ■ 5„, ■ • • is also summable, its sum being likewise s. For, T ^1 + ^2 + • • • + ^"+1 _ J __£ 1_ I T ^1 + • • • + 5n+l »=. n + I ~ n=« W + I n=» M + I _ J JQ + 5i + • • • + ^n+1 J -yp + 5i + • • • + ^n+1 W + 2 ~»=» «+I ~„=. M+2 n+i = Li 1 = S. * By this theorem requirement (iii) is satisfied. 8 UNIVERSITY OF MISSOURI STUDIES To return now to Theorem D; we wish to prove that if Mo + Ml + M2 + • • • is summable to 5, then Mi + Mz + • ■ • is summable to 5 — wo. The sequence corresponding to the series uo + Ui + ut + ■ • • is Uo, Uo + Ui, • • • . By the lemma proved above, it follows that the sequence mo+mi, Mo+Mi+Mz. ■ • • or Si, 52, • • • is summable to s. The sequence corresponding to Ml + M2 + • • • is Ml, Ml + M2, • • • which may be written Si — Mo, 52 — Mo, • • •. Now -J- r (5i - Mo) + (52 - Mo) -I + (5n - Mp) "j n=» L « J T /'5i +52 + ••• +5„ \ = JLi 1 Mo I = 5 — Mo. Theorem e:* // I «0 + Ml + • • • \ vn + vi + ■ ■ ■ are summable to u and v respectively, then the series (mo + Vo) + {ui + vi) + • ■ • is summable to u + v. Writing Sn = Uo + Ui + • ■ • + Un, t„ = Vo + Vi + ■ ■ • + v„, we have 5„ + /„ = (mo + Do) + (mi + »i) + • • ■ + (m„ + t>„). We obtain: J {so + to) + (Sl + tl) + ■■■ + (5„ + tn) « + I X -Jo + 5i + • • • + 5„ X 'O + <1 + ■•■ + tn , = Li -T- h Li z~r~r = U + V. Cesciro's definition of summability has accordingly been justified from the theoretical standpoint of our requirements for any generalized definition. We may naturally ask the practical question: how large is the class of series with which this defi- nition enables us to deal? A partial answer to this question is contained in the following proposition: * By this theorem requirement (iv) is satisfied. See also note p. 19. DEFINITION OF SUM OF A DIVERGENT SERIES 9 Theorem f: A necessary condition for the summability of the series uq + Ui + •■• + u„ • ■ ■ is — = O. ru=ao n Since the series is summable, J J0 + 5l+ ••• +J„_i S0 + S1+ ■■■ +S n J-J ~ JL ; = o = Xi ^'' + ^' + • • • + ^"-1 _ T ^0 + ^1 + • • • + 5. »=« w . — n Hence : LSn n=«> n ^7 = -L* Z = Lt ~ -Li — = 0. n=« n n-„ n „=„ M „=„ n We are accordingly limited to series for which (I) L- = o. n=ao n But such a simple series as 1— 2 + 3 — 4 + 5-- fails to satisfy this condition. Furthermore, this series can be easily evaluated by following out the principle of Euler; for if we put 2C = I in the expansion: = 1 - 2x + yc^---, (I + xY we obtain i = i-2+3-4+--. We are thus led to extend, with Ces^ro, the above definition of summability of order i, to summability of order 2. We say that a series is summable of order 2, if J (« + l)5o + nSt + ■ ■ • + 25n-l + ^n _ lO UNIVERSITY OF MISSOURI STUDIES A necessary condition* for the existence of this limit is that SO that we cannot evaluate the series, Kr + i) rjr + i)(r + 2) '"'+ 2! " 3! ^ ■ ' although we obtain by Euler's method, (1+^) and accordingly I , rjr + i) r{r + iKM- 2) ^ = i-^+^n"- — ^! — +•■■• We are thus led to state the following more general definition : Definition :t The series Mo + «i + "2 + • • • is summable of order r, if r is the smallest integer for which there exists the limit: r(r+i)---(r+w-i) r(r+i)- • •(r+w-2) ^^ ;;! +^' («-!)! ■•■■■■ r(r+i) , ^""^ »t (r+i)(r+2)---(r+«) " n\ This definition includes convergence for r = o ; it also includes the other definitions given above for r = i, 2 respectively. We shall not prove that this definition satisfies the requirements of page 2; this is easily verified. J Let us now return to Cesiro's first definition, and observe that we may generalize it in a more natural way. * Bromwich, loc. cit., p. 318. t Cesiro, loc. cit. t This is done in a more general case, infra, pp. 55-57. DEFINITION OF SUM OF A DIVERGENT SERIES II Definition:* Let io + 5i + • • • + Sn (3) / (i) = n + I / (r+l) = y 33 J 2, M + I then the smallest integer r for which Ij tJ-''> exists, shall make the series summable of order r. To distinguish this definition from that on page lo, we shall call the definitions Ces^ro-summability of order r and Holder- summability of order r, denoting them briefly by (Cr) and (ff,) respectively. It is knownf that these two definitions are equiv- alent for the same r. We may now ask how big a class of series this generalized definition enables us to deal with. If a series is (Cr), thenj Accordingly the series i— < + /* — /*+••• ('>i) does not have a sum (C) for any value of r; since L^ + o. t>i. We are thus led to generalize still further the definition for the sum of a series. From the definition given on page lo, it is clear that we may write Cesiro's forms as follows: * ~ »=« L Co + Ol + • • • + On J ' * Holder: Mathentatische Annalen, Bd. 20, p. 535. t Schnee: Math. Annalen, Vol. LXVII (1909). P- "O- Ford: Am. Journal of Math., Vol. XXXII (1909). P- 3I5- t Borel, Series divcrgentes, p. 92. 12 UNIVERSITY OF MISSOURI STUDIES where the a* are functions of both n and r, r being fixed.* Let us choose as our definitionf _ -J- J r go(r)iro + ai(y)5i + • • • + OnW^n "] "" " rti il L ao(r) + ai(r) + ■ • • + c„(r) J ' In particular we shall take ap{r) = r^lp\, and obtain (4) 5= L L r r r" 5o + 5i-+52^+ ••• +^n^ r r i+- + -,+ --- + n\ r=. giving for its sum s + Mo. The converse, however, is not necessarily true. Thus if the series mq + Mi + M2 + • • ■ is summable by (5), it does not followj that the series Mi + M2 + • • • is summable by (5). Since this fact is opposed to the requirement (iii), page 2, we are led to modify the above integral definition, and to state, with Borel, the following generalization: Definition: The series mo + Mi + M2 + • • • shall be called ab- solutely summable, whenever the integrals | e~' | u{r) \ dr, r ...... ^... .... .J: ... .. derivative. That this definition satisfies requirement (ni) is proved by the following theorem : § Theorem h: If either of the series I Mo + Ml + M2 + • • • 1 Ml + M2 + • • • is absolutely summable, so is the other; and if s, t be their respec- tive values, we have s — Uo = t. We shall not enter into the further generalizations which have been given by Borel himself and by Le Roy.|| * Borel, loc. cit., p. loi. t We shall call the two definitions given by Borel, the Borel-mean and the Borel-integral definition respectively. t For an example, see Hardy, Quarterly Journal, Vol. 35 (1903), p. 30. § Borel, loc. cit. II Le Roy: Annates de la FacuUe de Sciences de Toulouse (2° series), t. 2 (1902), p. 317. See p. 60, footnote. § 3- AVERAGEABLE SEQUENCES On page 4 we have considered the series j I - 1 + I - I + ••■ li+o-i + i+o-i + ---, and, replacing them by their respective sequences, we obtained U = I, o, I, o, ••• 1 f = I, I, o, I, I, o, ■••. The probability-method of Leibniz* consists in taking for the sum of the sequence, the average of its limit-values. This method has been justified by the theorems of Frobeniusf and Cesiro,t and the further generalizations. We propose now to give a justification of the method from another point of view. To define the sum of a sequence as the average of its limit-values is obviously not adequate; for although we can tell that the limit I is to be counted twice in the sequence considered above, I, I, o, I, I, o, •••. it is not easy or even possible to state the multiplicity of the limit-values in general, as is evident from the following example: Si = So, Sl, S2, • • • Sn, * ■ 5 . = o, i + mM , \ n = O, I,: To meet this difficulty, we shall proceed as follows. Let us assume, to be concrete, § that the sequence So, Sl, $2, • • • s„, ■ • • * See page 3. t See page 4. t See page 5. § We shall go into every detail in only this simple case; the later general- izations we shall outline only briefly. 15 l6 UNIVERSITY OF MISSOURI STUDIES has two limit-values h and k. Then we have I Sm - h\ < e, \s„ - /2 1 < e, for an infinite number of values of m and of n, provided m, n>N. Having chosen e and N, let us now choose i> N; then there will be m of these i numbers 5,- which fall in the interval about h, and n which fall in the interval about h. Since m and n are func- tions of i, we may write m = fi{i), n = fi{i). If we choose e suffi- ciently small, and i > N, we shall have fiii)+f2ii) +k=i, where fe is a constant independent of i. Definition : The sequence Sc, Si, Si, • • • Sn, • • ■ , having h and h as limit-values, shall be called averageable and have s for its sum provided J r /i(i)/i+/2(i)/2 l itil Mi)+Mi) J '• That this limit, when it exists, does not depend upon the particular e we have chosen follows at once. For if we take e wi+gii(w)w2 "| _ ilL /i(«)+/2(n) J'^iiL gi(n)+g2(«) J '^'- Thus it is seen that the requirements* of page 2 are satisfied by our definition. The extension of the definition to the case of sequences with any finite number of Hmit values is obvious. Definition: A sequence having k limit values, U, h, • • ■ h, shall be called averageable, and have s for its value, if T.fn{i)ln n=l Z/n« = 5. It can be easily verified that Theorem i applies to this extended definition. But we can generalize the notion of averageability even to cases where the sequence has an infinite number of limit-values. Let us consider a reducible sequence, and let us write: (£) = (£<»>) = 5o, si, S2, ■■■ Sn, ■■■ (£»>) = /o«». /l»\ h^'\ ■ ■ • in"', • • • (£<«) = /o<«, /l"', /2«'. • • • /n®, • ■ • (£W) = V*', //*', ;2<". •••/„<*', •••. where the sequence (£'■''>) consists of the limit values of the sequence (£<'~"). Since the sequence is assumed to be reducible, there exists a k such that (£<*+») = 0. Then (£) is reducible of order k, and (£'*') has only a finite number of elements. * Requirement (v) is satisfied by each definition considered. 20 UNIVERSITY OF MISSOURI STUDIES Let us assume that our sequence is reducible of order k, and that (£(*>) has for its elements Zp^*\ /i'*>, • • • Z^^". If now we choose e sufficiently small, all but a finite number of the /i'*~'> will fall in the intervals | /,<*-" — /p^*> \ < e, p = o, i, 2, ■ ■ ■ p. Suppose that the finite number of /i^*~'' which do not fall in any of these intervals is p\, and call them, Wi^*~". OT2''~", • • • ?Wp/*~". We can choose Ci < e, so small that only a finite number, pi, of the /<<*"''' do not fall in any of the intervals above, or in the intervals | /<<*"''> — Wp'*"'' \ N, if e is the largest of the e„ |ii - Si' I < e, i = I, 2, ■■• fi{n, e) ' I ^2 - Si" I < e, i = 1,2, • • • fiin, e) I Ip-Si^"'' I < e, i = I, 2, • • • fpin, e) We have accordingly: KM 4-/2/2+ ••• +fpiv) - lisi' + where s.-^'^ are those j, which fall in the e-in- terval about /,-. + SfJ) + + isi^"^ + + ^/ <"')]! <(/i+/2--- +/>. Since (5/ + 52' + + Sf,d+ ■■■ +(5i<''>+---+V) where g=/i+/2+ ••• +/?. and m is sufficiently large, we have: 22 UNIVERSITY OF MISSOURI STUDIES \flh + fih + • • ■ + fpip Sm+l + Sm+i + • • • + Sm+g < e. Hence ■Sfn + •Jm+IT ■ • ■ "r ^m+g j J [ fih+hh + ---+fpU ^j [s «=.L /1+/2+ •■•+/p J ,=.L J=«L 2 J provided either limit exists. By Theorem 2, the left-hand limit exists independently of e; accordingly the right-hand limit exists; that is, the given sequence is summable (Ci). In practice, the following proposition, a corollary of the theorem just proved, will be found useful: Corollary: If for some positive integer k, and for every positive integer i < k, the sequence 5,-, 5,+*, 5.+2i, • • • converges, then the sequence Si, 52, • • • is summable (Ci). Let us take as an example the sequence 5i = t log ( I -|- T J , i odd = 0, i even to which it is not easy to apply the formula T ^1 + ^2 + • • • + ^f. n~« ri We see, however, that the two sequences S\, 5:!, • • ■ I 52, S4, • • • J converge; hence the given sequence is summable (Cj). § 4- PRODUCT DEFINITIONS In dealing directly with sequences, the Cauchy-product* of two series does not appear to be entirely natural. Even in the case of convergent sequences, a more natural definition of product is close to hand. In fact, if j and / are the respective sums of two convergent sequences, {^0, Sl, 52, • • • Sn, ■ • • 'oi h, ti, • ■ ■ t„, • • •, then it follows from a fundamental theorem of limits that Ij Sjn = St. We are accordingly ledf to propose the following Definition : The Tiatural-product of two sequences, So, Sl, S2, ■ ■ • Sn, • • • ; ^Oi 'li ' ' ' tn, ' ' • , is the sequence: Soto, Siti, ■ ■ ■ sJn, • ■ ■■ We may then state the obvious proposition: Theorem: The natural-product of two convergent sequences, whose values are s and t respectively, is itself convergent; and its value is st. If we compare this theorem with the corresponding theorem J for the Cauchy-product, it will be seen at once that the natural- product is of superior value to the Cauchy-product, in the case of convergent sequences of constant terms. In the case of sequences which are not convergent, however, the natural- product can play no part. For consider the simple example, * See page 5. t Baire: Cours D'analyse, t. i. X Theorem B, page 6. 23 24 UNIVERSITY OF MISSOURI STUDIES S = 1, O, I, O, ■•• / = I, O, I, O, •■ • W = I, O, I, O, ■ • -, where the sequence whose value is w is the natural-product of the two sequences whose values are s and t respectively. Here s = t = w = ^, and accordingly w + st. We are consequently led to generalize the definition for the product of two sequences. Let us consider again the two sequences f So, 5l, 52, • • • ini • • • Itf), tl, ti, • • ■ tn, • • • and let us form the array : Solo,] Sotu\ S^i,\ ■ ■ • S4n, ^ifo. Sl«l,j 5l/2,i ■ ■ ■ Sltn, 52/0, 52/1, J2i;2,| • ■ ■ Sltn, sJo, 5n/l, Snt2, ■ • • Sntn, . . . Definition: The sequence formed by following the successive lines which form squares with the boundaries of the array, i. e., Jo^o; Sati, S]tu S\ta\ SB,t2, Sit2, 52^2, s^ti, s^to; • ■ •, shall be called the square-product of the two sequences. We shall now prove the following theorem : Theorem 4: The square-product of two averageable sequences is averageable, and its value is equal to the product of their values. Let the given sequences be DEFINITION OF SUM OF A DIVERGENT SERIES 25 {5 = Sd, 5i, • • • S„, • • • I = to, ti, • • • tn, • • • ; we wish to prove that the sequence S(^o', Soil, Siti, s\io; Soti, Siti, Siti, Siti, s^to', • • ■ is averageable, and that its value is st. We shall assume* that the sequence (s) has the two limit-values h, h, and that the sequence (<) has the two limit-values mi, m^. The only limit- values of the product sequence are then: hmu hnii, hnii and hnii. We are given niiL Mn)+Mn) ]• and we wish to consider: )limi + Fi2{n)limi + FiiMhmi + Fn(n)l2m2~\ Fnin) + Fiiin) + F2M + F^^in) J' where Fij{n) is the number of elements of the product sequence near /,Wj. If we pick n elements from the product sequence, we observe: Fn{n) = /i(»)gi(«) + ifeii H Fii{n) = Mn)gi{n) + kn Fn{n) = /i(M)g!!(«) + kiiiX Fi2{n) = }7.{n)gi{n) -f k^, where k^ are constants independent of n. We have, accordingly, _ r Fii(w)/iWi + Fii{n)hmi -h F2x{n)hmi -\- F2i{n)hmi ~\ hi Fnin) + Fi.in) + Fn{n) + Fn(n) J l/i(«)gi(w) + kn]hmi + [fi{n)g2{n) + feul/iWj ^ J + [Mn)giin) + kii]hmi + [Mn)g2(n) + kulhrn j " /i(w)gi(n) + ku +Mn)gi{n) + kn +/2(w)gi(n) + hi +Mn)g2in) + *22 ' The proof for the general case is precisely similar. 26 UNIVERSITY OF MISSOURI STUDIES = L 'fi{n)gi{n)hmi + fi{n)gi{n)lim2 + fi{n)gi{n)hmi + Mn)g2(n)hm2 /i(«)gi(«) + Un)g2{n) + h{n)gi{n) + h{n)g2{n) J _ -J r [h{n)h +h{n)k] [gi(«)wi + g2(w)m2] 1 ^ ilL [/l(«)+/2(«)][glW+g2(«)] J For example, the square-product of the sequences f s = I, o, I, o, •■• 1/ = 1, o, I, o, •••, is w=i; o, o, o; i,o, i,o, i; o,o, o, o, o, o, o; i, o, i, 0, i, o, i,o; • ■ •. If we choose m terms of this sequence, and let (2n)^ be the largest square of an even integer less than or equal to m, so that m = {2nY + k, o < jfe < 8m + 4, we get : _j\ [i+3-l |-(2w-i)]i+[m-(i-| \-2n-\)]-o 1 n Li — = L n=«> Thus it is verified that w = s • t. Although it is true that the natural-product is better adapted to convergent sequences than the Cauchy-product, and that the square-product is better suited for averageable sequences, it must be remembered that in analysis the things that arise frequently are not sequences of constant terms, but rather series of variable terms, notably power series. In the case of power series, the Cauchy-product is certainly more valuable; for if we multiply two such series according to the Cauchy scheme, we obtain the same result which is given by multiplying the two series as if they were polynomials, thus: DEFINITION OF SUM OF A DIVERGENT SERIES 27 {u{x) = Mo + UiX + U2X^ + U3X^ + • • • + UnX" + • • • I v{x) == Vq + vix + ViX^ + fsa^ + • • • + t;„a;" + • • • W{x) =u{x)-v{x)=U(flo+{UoVi + UiVn)x+{uoVz+UiVi + UiVo)x^-\ . Furthermore, to this symbolic advantage is added the theoretical one which is contained in the following theorem, due to Ces^ro,* which is a generalization of Theorem B. Theorem (j): The Cauchy-producl of two Cesiro-summable series, of orders p and q, and of values s and t respectively, is itself Cesdro-summable of order at most /> + g + I, and its value is st. In certain special cases, we can slightly improve upon the results of Ces^ro's theorem. Thus, if two series are convergent (i. e., summable of order o), their product must be summable of order at most i. If, however, one of these series converges absolutely, then the product-series is convergent, f as has already been stated. J Similarly, the Cauchy-product of two Cesiro- summable series, one of order r, the other convergent, is sum- mable (Cr+i); if the convergent series happens to be absolutely convergent, however, the product can be shown to be summable (Cr). Theorem 5 : The Cauchy-product of a Cesiro-summable series of order r by an absolutely convergent series, is itself Cesiro-sum- mable of order r. Let Sn = UO + Ul + ■ ■ ■ + tin, tn = Va + Vl + ■■■ + Vn, Wn = Mof n + UlVn-l + • • • + Mn^'o, yn = Wo + Wl + • • ■ + Wn. * Ces4ro: Bull, des Sciences math., t. XIV, 1890. t Mertens, Journal de CreUe, t. 79, p. 182. % P. 5, supra. 28 UNIVERSITY OF MISSOURI STUDIES r{r+i) ■■■ (r + n - i) r{r + j)- ■ ■ (r + n - 2) (n- I)! + • • ■ + 3'n- r{r + I) + yu-i-r+yn, _ _ , r{r + 1) ■■■ (r + n- 1) r(r + i) ■ ■ ■ (r + n - 2) m! (w — i)! r(r + i) r(r + I) • • • (r + « - i) (''. ») = Ti ; tn = 2! r„ (r, n) ' We assume: Ij5„ = 5, L [|Wo| + |Mi| + • • ■ + \Un\ ] = ^, T n=« (r + x) • • • (r + n) = i. and we wish to prove : «=« (r + i) • • ■ (r + n) = 5-i. Proof: Lemma: If ■* n r„ - r„_ J-' 7 r Tn 7 I \ = 'i then Ju -, ; r 7-^ r = O. ,.=- (>•+ i) ••• (r + w) „=.(r + i) • ■ ■ (r 4- w) ' «! For L p = \,2,--- p. ^_n I n—p ir+i)---{r + n) ~ (r + i) . . . (r + «) n\ n\ DEFINITION OF SUM OF A DIVERGENT SERIES ■* n ■* n— p 89 n=« (r+i) ••• (r + n) {r + i) • ■• {r + n - p) n (n-p)l (r+i) ■■■ (r + n-p) {n -p)\ (r+l)--- (r+n) = L T.. {r+i)---{r+n) jr+i)- ■ -{r+n- p) n\ in-p)\ w(w — i) • • • (n — />+ i) (r + n)-(j'+n -!)••• (r.+ w -/) + !) = / — / • I = o, Now yn = U^O + (MoI'I + Mll'o) + («0f 2 + Mli'l + M2f o) + ' " " + (Mofn + Ml!'n-1 + • ■ • + Un-lVl + U„Vq) = Mo(j'0 + 1^1+ • • • +1*1.)+ Ml(j'0 + J*! + • • • + I'n-l) + • • • + M„_l(!;o + Vl) + Mnfo. yn = «0 = 0,1,2, ■2U + 5) + 2U+J3)''^">^= .••-R< ll«oH-|«.|+ ••• +l««ll J^ + {|««+i| + ••■ + |Mn| + ••• + |m2„|)5, if w > A^. Now choose g so large, that Iwafi 1+ • • • + \u2„\ < . „ , 2 > Q for a// «. Moreover, |mo1 + • • • + |ii,| < i4 for a« q. Thus Similarly A +B Yin n=«,(r + I, 2n) r y2n+l =„(r + I, 2W + l) = S-t. = S-t. The theorem is now proved. In the case of power series, then, both the symbolic advan- tage and the theoretical importance of Theorems j and 5 lead 32 UNIVERSITY OF MISSOURI STUDIES naturally to the Cauchy-product. This advantage does not ap- pear, however, in case of sequences which do not correspond to power series, — for example, in Fourier's series; in this case, the square-product may be of greater service than the Cauchy- product. We should observe, however, that while the square- product may justly replace the Cauchy definition of multipli- cation, in certain cases; the definition of averageability has the disadvantage of presupposing the knowledge of the limit- values ; and these are not always easy to determine even in the case of sequences of constant terms. § 5- ON CERTAIN POSSIBLE DEFINITIONS OF SUMMABILITY Cauchy has proved* the following theorem, which we shall show is equivalent to Theorem c. Theorem k: If u„> o and L ^^^ = /, then L «„"" = /. Let ■ = tn+i, Wo — I, then Wn = tit2 • • ■ tn- Accordingly, whenever Ij /„ = ^ then L (/i<2 • • ■ t„y'" = /, provided /„ > o; and the last equation may be written T f log h+\o gh+ ■■■ + log tn \ '°^'=iil n )■ And if we finally write log /„ = 5„, we obtain the result that T Jl + ^2 + • • • + S„ Li = ■s n whenever 1j s„ = s. This statement is, however, precisely Theorem c. We see accordingly that Theorems c and k are equivalent, by means of the substitution * Cours d' Analyse: Oeuvres de Cauchy (2° serie), Vol. 3, pt. 3. 33 34 UNIVERSITY OF MISSOURI STUDIES — In+l — e U„ Let us make the further substitution 5„ = r„^„, and observe that the variables 5 „ and ?■„ on each side of this equation approach the same limit, provided L ^n = I- n=ao We may accordingly replace Theorem c, which we have just obtained again, by the following theorem: Theorem 6: // Tj r„ = r, and Tj » = I not monotonically, follows from the example: r„ = (- i)»+'log«, (P„ = I + (- i)»+' p^— , n+i, v>i = I. log n Here ii =0, L. ^„ = I n= 00 77 n= oD wo/ monotonically; j^ yiri + • • • + 'a + • • • + '2 + • • • + ynyn «=» SPl + V2 + • • • + Vn We may now apply Theorem L directly, by identifying monotonically. We first suppose that / =1= o, and form the sequence t„/t, so that T ^'■ monotonically. Accordingly, by Theorem 7, tl , t2 1 'n s^-+S2j +...+s„j Li = s ^ Siti + Siti + • ■ • + sj„ Li = St. DEFINITION OF SUM OF A DIVERGENT SERIES 37 If / = o, we form the sequence i + /„, so that L(i +/J = I monotonically; consequently, by Theorem 7, J ^ L ^'( ^ + ^') + ^'(l + <2) + • • • + 5„(l + „ so as to make the two definitions equivalent; and we may state the following theorem : Theorem 9; // IjiPn = I n=<» monotonically, then whenever either of the two definitions — n, i- e., LSiVi + Si'ipi + • ■ • + s„^„ = s. n This amounts to saying that the sequence {s„» = I monotonically, and _ J r ^iyiyi + Si — ,i = 2,3, ■■•. so that Siipi = o Si »=« we shall assume more generally that (9) is satisfied, and take as our definition. 42 (lo) UNIVERSITY OF MISSOURI STUDIES 'Pin) 5,^(I)[^(I) + ^ +S2(p(2) ^ — + ■■■ + + ••• + n n + ■ +Sn(p{n) m r = -y. I. If ¥>(«) — ^(«) = I, we obtain: = L M J X r , Si + 52 , 5i + 52 + 53 , 5i + 52 + S„ J-j J 5i H — 1- :: 1- • ■ • + - n T [ h + h+-J_-+tn~\ , ^ 5, = ±j where /„ = — »=« L " J which is Holder summability of order 2. If + 52 + • ■ • + 5„ we obtain : Si2 -f unctions equal to unity, WQ shall obtain all of Holder's forms; while by a suitable choice of these />-functions, all of the Ces^ro-forms might also be ob- tained. But though the process is quite clearly defined, the algebraic details become too complicated to carry this work any further. The fact, however, that we may use, as a definition of summability, the limit of an expression in which the coefficients of the Si are not specifically named, but are given in terms of functions satisfying certain conditions, suggests a more general view of summability, which we shall proceed to develop in the next article. § 6. DEFINITIONS OF EVALUABILITY We have now considered a large number of definitions of summability. It is natural to ask whether all those definitions do not have some common properties. Excepting for the moment Borel's definitions, to which we shall return later, we can say that all* the definitions of summability which we have considered have the following properties in common: If ai{n) represents the coefficient of Si in any of the expressions whose limit gives rise to one of the definitions of summability, then: (i) L ai{n) = o, for fixed i, 71=00 (ii) L [ai(w) + ai{n) + • • • + a„(w)] = i, n=oo (iii) ai{n) > t o for all i and n. That properties (i) and (iii) are common to all* of the definitions under consideration is easily verified. We proceed to show that the same is true of property (ii). Beginning with Ces^ro- summability of order r, we shall show that the sum of the coef- ficients of the numerator, divided by the denominator, is iden- tically equal to unity. For this purpose we write: (i - »;)-<'■+» = {i +X + X' + x^ ■■■ +X'' -\ )(i - x)-'. Equating the coefficients of ac" on each side of this identity, we obtain : ir+i){r + 2) ■■■ (r + n) , , r{r + i) , nl -i-fr-f-^j— +... r(r + i) ■ ■ • (r + n - i) * We exclude also definition (lo). t The equality sign occurs in the case of convergence. 46 DEFINITION OF SUM OF A DIVERGENT SERIES 47 SO that: r(r+i)(r + 2)---(r + n- i) . . r{r + i) , n! + ••• + 2; +r + i ^ (f + i)(r + 2) • • - (r+ w) ~^- m! Turning now to Holder's definitions, we observe that for order i, the sum of the coefficients of the Si is identically equal to unity — this being in fact a special case of the case just con- sidered. Suppose now that hi, hi, • • -hn are the coefficients of Holder's definition of order p, so that L [h\Si -f ^252 + • • • + hnS„] = 5. If we assume that hi + hi + • ■ • + hn = i for order p, we obtain for order ^ + i, putting _ h + ti + ■ • • + t„ n n and the sum of the coefficients becomes = hi + hi + ■■• + h„ ^ I. Thus the proof of (ii) for Holder's definitions is completed by mathematical induction. Let us now consider formula (7). We shall show that 48 UNIVERSITY OF MISSOURI STUDIES where n , n n log - + log - + • • • + log w„ = If then Hence Accordingly, n n n ^ \ I 2 n — \ ) I 2 « - - I (« I)! I'n+1 Vn = (■ -^) n L !-„"" = L ^n+1 _, e. « = « n=oo Vn L «„ = Liogf„i'" = I. Finally since we have assumed in the ^-definition that L if,{n) = I, 71=00 it follows that J y>(l) + o, (iv) L [ai(n)5i + ai{n)s2 + • • ■ + a„{n)s„] = s. We shall now justify this definition by proving the following theorem : Theorem ii : If a series is convergent then it is A-evaluable* By (iv) we may write: (v) ■ [ai{n) + ai{n) + • • • + fl„(n)] + r„ = i, Li /"n = o. Now, by (v), I ai{n)si + ai{n)si + • • • + a„(w)5„ — s | = I {ai{n)si + ai{n)s2 + • • • + an{n)sn\ - (ai(«) + osW + • • • + an(w) + rn)s \ < 1 ai{n){si - s) + ai{n){s2 - s) -\ + ap{n)isp - s) \ + I ap+iin) (sp+i - 5) + • • • + c„(«) (s„ - s) 1 + I f„s I . Since the series is convergent, we can choose i so large that \Si - 5| < 7;, i> p. Let / be the largest of the numbers | -r< - s |, for i = i, 2, • • • />. We have, then, I Oi(«)si + 02(«)52 + • • • + a„(n)s„ - s \ < {ai(M) 1 5i - 5 I + • • ■ + Op(n) I 5p - -si) • Theorem 1 1 obtains if condition (iii) is replaced by the broader condition; I Oi(w) I + I a2(») I + • • • + I an(n) | < K. 50 UNIVERSITY OF MISSOURI STUDIES + {ap+i{n)\s^i - s\ + ■■■ + a„(«)|5„ - 5|! + \r„s\ < {ai(»)H \-aj,{n)}l+ {ap+i{n)-\ ]ra„{n)} T]+\rns\ <5l + v+\rns\, n> N* e e e e e <^ + - + -by(v). if^=-^. , = ^. e. Hence ~Lt lai{n)si + ai{n)Si + ■ ■ ■ + a„(«)s„] = s. Our definition of i4-evaluability is now justified. The question naturally suggests itself as to whether for a sequence (s„) which diverges to + *, n L 2 ai{n)si = + 00 . The answer, which is in the affirmative, is embodied in the fol- lowing theorem : Theorem iia: // IjS„ = + 00, n=QO and conditions (i), (ii), (iii) are satisfied, then n Li '^ai{n)Si = + 00. 71=00 i~\ By hypothesis, Sn > N^ n > m. Hence n m n o-„ = X (^i{n)Si = ]C a-i^Si + X) aiMsi 1=1 1=1 m+l m n >'^ai{n)si + N 5^ a.(w). * By (i), [a,(n) + • • • + Op(»)] < S, n > N, p having been chosen first, and then held fast. By (iii), [oj,+i(w) + ■ ■ • + a„(n)] < [a,{,n) -\ + o„(m)] < I by (ii). DEFINITION OF SUM OF A DIVERGENT SERIES 51 Since [m 71-1 T.ai{n)si + N T, Oiin) = N, it follows that Minimum Ij o-„ > iV; and since N is an arbitrary number, Lffn = + 00. n=oo We have seen that the generalized definition includes a large number of the specific definitions of summability which we have considered. But we see now that if we take any functions whatever for ai{n), subject merely to the restrictions (i), (ii) and (iii), we may obtain a possible definition of summability. Thus, we may take as our definition, for example. (II) s=Li 2 • + log n = L S1+-S2+ ■ ■+^. ■+;+■ •■+^ This formula is of interest to us, since it affords an example of a definition which is broader than Ces^ro-summability of order i , and yet perhaps not so general as that of order 2. For since i/« steadily decreases, it follows from Theorem 8 that formula (11) gives a value to all series that are Ces^ro-summable of order i , and that these values are the same for both definitions. That (11) is really more general than summability of order I follows from the example i -3 + 5-7+ •••• This series is not summable of order i , since however we obtain from (11), for the corresponding sequence, 5„ = (- !)"+'«, 52 UNIVERSITY OF MISSOURI STUDIES [' - I + I J »=«. L log M J logn Nevertheless, (ii) is probably not equivalent to summability of order 2, as the following reasoning suggests. A necessary condition that a series give a result by (ii) is n=:« w log K This is not, however, a necessary condition for summability of order sf — so that we might find a series for which «=»Mlogw + 0, which is nevertheless summable of order 2. We have seen that the ^-definition includes most of the cases of summability which we have discussed, but we have been obliged to omit Borel's definitions. In order to include the Borel- mean-definition, we shall now generalize Theorem 1 1 , as well as the definition which we have based upon it. Replacing aj(w) by ai{a), where a may be independent of n, Theorem (ii) may be stated in a more general form: Theorem 12: From the conditions: = Ij — ii- n=aon log. '0= Tj 71=00 [ '+j +•••+„;, '+^+--+-n J Li -j = Li i = 0. n=ooKiOgm n=» WlOgn t A necessary condition for summability of order 2 is Li -r = 0. See p. 10. DEFINITION OF SUM OF A DIVERGENT SERIES (i) li ai{a) = o for fixed i, 0=00 (ii) L [ai{a) + 02(0) + • • • + a„(a)] = i, n=oo (iii) a,(a) > o, (iv) Ju Sn = s, 53 vtay be deduced the result: Li Li [ai{a)si + a2{a)Si + a=cx) n=ao + a„{a)s„] = 5. We shall first show that L [aiiajsi + 02(0)52 + • • ■ + a„{a)Sn] exists for every definite a. Taking a definite value of a, \a„{a)sn + a„+i{a)sn+i + • • ■ + a„+p{a)Sn+p\ < a„{a)\Sn\ + ■ • • + an+p{a)\Sn+p\ N.anyp)) = e. Hence 2J fflnW^n converges for every value of a. Since eo 22o„(a)5„ n=l has a sense, we may write : Z^Clniod^n ~ 2^ a„{a)s„ — 2I a»(a) by (ii) 54 UNIVERSITY OF MISSOURI STUDIES = Sa„(o:)(5„-5) n=I m-l < hT, an(a) + e. < 2^ a„ia)isn - s) + 2 an{a){s„-s) since \s„ — s\ < e, n > m, and \s„ — s\ < H, n < m by (iv). Since, however, Ij S anW = o by (i), it follows that: Maximum Ij ^an{a)Sn — s o for a > o, it follows that (iii) is fulfilled. We might accordingly generalize our definition of evaluability, to include Borel's mean-definition, by using the hypotheses (i) to (iii) of Theorem I2 as a basis. It turns out, however, that we may generalize Theorem I2 still further, and that we can accordingly obtain a still more general definition of evaluability. DEFINITION OF SUM OF A DIVERGENT SERIES 55 Let US take as coefficients of the Si functions of both n and a, and write: (i) Li ai{a, n) = o, n=ao 71 (ii) li S diiot, n) = 1, . (iii) ai{a, n) > o. If now these conditions are fulfilled for a fixed value of a, and if Ij ^n = 5, n=eo it follows from Theorem ii, that n Jj 51 ai{a, n)si = 5. n=oo i—0 Since this limit exists for every value of a, under our hypothesis, we may write: TO (iv) Ij Ij S ciiia, n)si = s, a=ao n^oo i=0 and a definition that readily suggests itself, even when the series is not convergent, is that conditions (i) to (iv) be fulfilled. We have demanded at the very start, however, that every definition should satisfy certain fundamental requirements, which we have enumerated on page 2, and while the definition proposed does fulfil the first two of those requirements, as we have just seen, it does not fulfil the third requirement* without further restrictions on the coefficients. t Our third fundamental demand was that when the series Mo + «i + M2 + • • • + Mn + • • • has the value 5, then the series Mi + Mj + • • • + "n + • • • must have the value s — Mo; * The same is true, of course, for the ^-definition; we have deferred the similar considerations for that case, since they may be included under this more general one. t It is obvious that the fourth and fifth requirements are also fulfilled. 56 UNIVERSITY OF MISSOURI STUDIES or stated in terms of sequences, if s„ = Mo + "i + • • • + «n, when the sequence 5o, Si, Si, • • • 5„, • • • has the value s, then the sequence Si — mo, S2 — Uo, ■ • • s„ — uo, • • • has the value 5 — Wo. If we assume, for the moment, that whenever either one of the two sequences So, Si, Si, • ■ • 5„, • • • 5l> ^2, ■ * ■ ^ni * ■ ■ has the value s, the other does also; then we shall satisfy our third requirement if we prove that whenever Si, S2, S3, • • ■ Sn, • ■ • has the value 5, then Si — uo, S2 — «o, ^3 — "oi • • ■ Sn — ua, • • • has the value 5 — Uq. Now this it is easy to prove. For we have by iv, p. 55, n n Li L S ctt(«. w)(5i — Wo)= Ij ~Liz2o'iioc,n)si~UQ = s — Uo a=OD 71=00 i=-Q a^co n=QO i=s.O by (ii).p. 55- It remains then to consider under what restrictions we can justify our assumption that the two sequences So, Si, Si, • • • Sn, • • • ^ii ^2> ■ ■ ■ ^Bi ■ ■ ■ always have a value together. To get an idea as to the nature of the condition which we shall have to add, let us consider, for concreteness, what happens in the case of Borel's mean-definition. Using the notation of page 12, we have: a a^ a" s{cx) = So + Si- + Si--,+ ■ ■■ + J„ — j + • • • , I 2 ! n i s'{a) = Si + Si-+ ■■■ + Sn . _ ., + • • •, s'{a) - s{a) = Ml + Ms- + M3— , + • • • + M„ . _ ., + ■ • •. Borel's definition being DEFINITION OF SUM OF A DIVERGENT SERIES 57 If we assume* that Ij 5(a) = 00, a=go we have an indeterminate form, so that or Li s'ja) - s(a) «=» - =0, which may be written, Ije-" Ml + W2-+ J/3— + ... +M — = o. a=oo n=to l_ I 2 ! W ! J It is accordingly suggested that we assume, in general, (v) Li L/ lao(a, n)ui + Oi(a, m)M2 + • • • + c„(a, w)m„+i] = o. a=:ao n=QO As a matter of fact, this condition is sufficient,! for, from (iv) (iv) ~Li Ij [ao(a, w).So + ai(a, n)5i + • • • + a„(a, n)5„] = 5 and adding (iv) and (v) we obtain L Ij[ao(Q!, w)5i + Ci(q:, m)j2 + • • • + a„(Q:, «)i„+i] = J, a=ao n=oo which proves that when the sequence So, ^i, • • • Sn, • • • is eval- uable to s, so is the sequence Si, Si, ■ ■ • Sn, ■ • • ■ By subtracting (v) from the last limit we show in the same way that when the sequence Si, Sa, • • ■ s„, • • • is evaluable to s, so is the sequence ^0, Su Si, • • • s„, • • • . Thus, condition (v) causes our definition to satisfy the third requirement of page 2. If we wish to be able to drop any finite number of terms, we shall have to require a condition more genercd than (v), as we shall do in the following definition : * This assumption is not essential, since our object is simply to arrive at a certain condition on the ai(a, n). t Condition (v) is not satisfactory since it is a condition on the sequence, as well as on Oiin, a). It would be desirable to have on ai(n, a) further re- strictions, sufficient to cause (v) to hold for all sequences. 58 UNIVERSITY OF MISSOURI STUDIES Definition: A series shall be said to be B-evaltcable and to have the sum s whenever the following conditions are fulfilled: (i) Li ai(a, n) = o, n (ii)Ij ^ai(a, n) = I, n=co i—0 (iii) ai(a, n) ^ o, n (iv) L Lj S fliCa. «)*•• = ■s. a~aQ n=GO i=:0 n (v) 'Ll L SfliCa, M)M,+t = O, k = 1,2, ■■■ p. We have seen that this definition includes all of the definitions of summability which we have considered, except possibly the Borel-integral definition. We have not yet subjected this integral definition to the test of our fundamental requirements; let us now do this. That requirements (i) and (ii) are satisfied follows from the following theorem:* If J_j 5„ = s, n— 00 then I/O e~'u{r)dr = 5, where u{r) = Mo + Ml - + M2 — j + It is obvious, too, that requirements (iv) and (v) are satisfied. Let us accordingly limit our considerations to requirement (iii) . With regard to this requirement we have the following state of affairs :t * Hardy: Quarterly Journal, Vol. 35, p. 22; Bromwich, loc. cit., p. 269. t The quotation is taken from Bromwich, loc. cit., p. 271. The first of the propositions was proved by Borel, loc. cit., p. loi; Hardy proved the second proposition by an example: Quarterly Journal, Vol. 35 (1903), p. 30. DEFINITION OF SUM OF A DIVERGENT SERIES 59 "Any finite number of terms may be prefixed to a summable series, and the series will remain summable. . . . But the removal of even a single term from the beginning of the series may destroy the property of summability." Inasmuch then as the integral-definition fails to satisfy one of our fundamental requirements, we are obliged to rule it out. In fact Borel himself ruled it out,* replacing it by absolute summability. '\ This definition does satisfy requirement (iii), as Borel proves, t and it obviously satisfies requirements (ii), (iv) and (v). Furthermore, Borel makes the statementj that convergent series are always absolutely summable. Hence it would follow that the definition of absolute summability is to be retained, since it seems to satisfy all of the fundamental requirements. But Borel's statement that convergent series are always absolutely summable, is incorrect, as Hardy § has shown by the following example: (-1)' n =^^ Un = 0, n not a square In fact the series in question: -i+o+o+|+o+o+o+o-§+ is convergent, while e-'\u{r)\dr r is divergent. Thus, since absolute summability fails to satisfy * Loc. cit., p. 99. t See p. 14. J Loc. cit., p. 100. § Hardy, loc. cit. 6o UNIVERSITY OF MISSOURI STUDIES the first fundamental requirement, this definition too cannot be retained.* We have seen that the B-definition satisfies all of our funda- mental requirements, and that it includes as special cases all of the proposed definitions of summability which satisfy those requirements. Our definition of 5-summability is accordingly justified. We proceed to the statement of the following definitions: Definition i : A series shall be called abstractly-evaluable, and to have the value s, if the following conditions are fulfilled: (a) Li [ai{n)si + ai{n)s2 + • • • + c„(w)5j = s, {b) the fundamental requirements of page 2 are satisfied. Definition 2: An abstractly-evaluable series of functions of a variable shall be called uniformly evaltmble, if: L [ai{n)si{x) + a2{n)s2{x) + • • • + a„{n)sn{x)] ^^ = L fix, n) = s{x) uniformly. From these definitions follow at once several theorems. Theorem 13: A uniformly evaluable series of continuous functions represents a continuous function. '\ For /(*:, n) = ai{n)si{x) + - • • + an{n)sn{x) is a continuous function of x; and since Ij/(x, «) = s{x) uniformly, it follows that s{x) is continuous. Similarly, we should obtain in the usual way, the following two propositions: * It is for this reason that we omit from further considerations the integral definition and the extended definitions given by Borel himself and by Le Roy. See p. 14, supra. t The same proof applies when the continuity is with respect to some assemblage. DEFINITION OF SUM OF A DIVERGENT SERIES 6 1 Theorem 13A: ^ sufficient condition that an abstractly-evaluable series of continuous functions represent a continuous function is that the related sequence, f{x, n), have Dini's simple-uniform con- vergence.* Theorem 13B: A necessary and sufficient condition that an ab- stractly-evaluable series of continuous functions define a continuous function is that f{x, n) have ArzeWs quasi-uniform convergence.^ Theorem 14: A uniformly evaluable series of continuous functions may be integrated term by term. We wish to prove in this case that I L [aiin)si{x) + ai{n)si{x) + • • • + a„in)s„{x)]dx = L I [ai(w)si(x) + a^{n)s^{x) + • • • + a„(w)s „(*;)] da; n=QO t/a or rlj/(x, w)dx = L I fix,n)dx, n=ao n=Go t/o but this equation is precisely a statement of the theorem that a uniformly convergent sequence of continuous functions may be integrated term by term. Theorem 15: If a series of continuous functions is convergent for all values of x in an interval, except possibly for x = xo; and if two sets of functions 0,(71), bi{n) render the series abstractly- evaluable at xo, to the values s and t respectively; then, if the evalua- bility of each of the definitions is uniform in the interval, then s = t. Letting n fix, n) = ^ai{n)Si{x), and n g{x, n) = J^bi{n)Si{x), * Dini: Fundamenli per la leoretica delle Funzioni di variabili reali. Pise, 1878, p. 103. t Arzeli: Memoir es de Bologne, 1899. 62 UNIVERSITY OF MISSOURI STUDIES and remembering that since the series is convergent, x =H xo, it is true that Tjfix, n) = L/g(:ic, w), X + xo, n=ao n=GO we have from the uniformity, Ij Jjf(x, n) = Jjfixo, n) = sA x=zq n= 00 n= oo I L Jj g{x, «) = Li g{xB, n) =,t \ and hence s = t. We may obviously state the preceding theorem in the following more general manner: Theorem 15A: If a series of functions continuous on an as- semblage (E) is convergent at all points of (£), except possibly at X = a;o, which is a limit point of (E); and if two sets of functions cti{n), bi{n) render the series abstractly-evaluable at xq, to the values s and t respectively; then, if the evaluability of each of the definitions is uniform on (£), it follows that s = t. § 7- APPLICATIONS We shall first consider an application of the definition of abstract evaluability to integral equations, and we shall obtain a generalization* of a theorem due to Volterra.f Let us seek for a continuous solution of the integral equation, u{x) =f{x)+ f K{x, Ouii)dt where K{x, y) is continuous, t ta<.x(»:) exist, in the interval (a, a + h), that § (i6) R„ = -/"(a + eh), o < < I. * Math. Annalen, Bd. 58, 1904, p. 51. t/(3c) may become infinite at a finite number of points. t Transactions, Am. Math. Soc., Vol. 10 (1909), p. 391. § This is Lagrange's form for the remainder. See Goursat-Hedrick, loc. cit., p. 90. (17) /(a + h)= f{a) + hfia) + -J"{a) + if and only if A" n\- DEFINITION OF SUM OF A DIVERGENT SERIES 67 From (15) it is obvious that 2! 71 = 00 If it should turn out that Li i?„ = fe + o, 71=00 then it follows that the series of the right member of (17) cannot represent f{a + h). But if L i?„ does not exist, though the 71=00 series cannot then be convergent, it may be possible to choose a definition of sum which will give for its value f{a + h). Thus we obtain from (15) and (16) -J2si=f{a + h)--'ZRi= Ka + h) - R„, n 1=1 n i=i (18) -I Ri = ^ rHa + e,)h, I " - Rn = ~ /Li Ri- n i=\ As before, we consider three possibilities. If L i?n = o, n=oo then IjlZsi=fia + h); 71=00 " i=l if L. i?n = /fe + o, L - E 5, + /(a + /?); 7t=:00 71=00 ^ i=l I " and if L Rn does not exist, Li - £ 5, does not exist. 71=00 n=oo '^ i=l This result is not satisfactory as it stands, however, because of the Oi which appear in (18), and which may differ with i. 68 UNIVERSITY OF MISSOURI STUDIES We shall accordingly proceed to obtain another form for R„ We have: nf(a) + in- i)/'(a) ^ + (n-2)f"{a) ^ + • • • +'2f"-'Ka) T- -7-, +/"-'(«) I ^ ' ■■' ^"■^(w-2)! ' •' ^^(«-i)! - Z^Si = . w " n For fixed a and fi we let the difference /(a + fe) --T,Si =-tP = Rn, n i=i p and we consider the auxiliary function ^{x) = ^ I nf{a + h)- [«/(x) + (« - i)"^^^^^^^/'W Since ^(a) = ^(c -\- h) = o, it follows that tp'(a + 6A) = o, o < e < I. But n^'{x) = - [n/'(*) + (n - I) ^^-t||-=^V"W + (« - 2) — ^1 — r"{x) + • • • + („-i)! /"(^)J + [(« - i)nx) + {n-2) ^°^f,~''V "W + (« - 3) ('^ + ^-^)y >(-,) +... + („_,. ^ -;c)--'np] [ /■(.) + <-i±^/"M + ^i±|p^V" W + ■ + |w-l)! ^"^^^ - (« + A - ^)'-'«P J . DEFINITION OF SUM OF A DIVERGENT SERIES 69 Since i(>'{a+eh) = o, ^ = ^. (/'(?) + 7i/"(«) + ^/'"(f) + • • • where ^ = a+ eh, a + h - ^ = hii - e) = \, o < e < I. If we choose /> = i, we obtain: ^"^ = ^^ = l{^'^^^ +T/"(«) +^/"'(« + •■• (20) \n-2 If now then if and only if I " -H Si = f{a + h) - R„, L ^ E Si = /(a + A) 71=CD " 1=1 Ij i?„ = O. We have thus proved: Theorem it. If the first n derivatives of f{x) exist in the interval (a, a + h), then h fc2 },n /(« + h)= f{a) + 7j /'(a) + -,/"(a) + • • • + -,/"(a) + w//ere /Ac infinite series is Cesdro-summable of order i, provided L i?n = o, n=oo 70 UNIVERSITY OF MISSOURI STUDIES ^ = a + eh, \ = a + h — ^, o < 6 < i. Turning now to the (^-definition,* n S » = I. 71=00 DEFINITION OF SUM OF A DIVERGENT SERIES 7 1 and we construct the function: ,) which gives the series a finite value Sr, then the series converges absolutely. We first observe that we may assume the series to have an infinite number of terms of each sign; for otherwise, the theorem * Here even requirement (l) is not fulfilled; see p. 56. t We proved the satisfaction of the first requirement in all our cases except Borel's absolute summability; similar proofs can be given for the second re- quirement, some of which are included in Theorem i la. 74 UNIVERSITY OF MISSOURI STUDIES is proved, since the series cannot in that case diverge unless it diverge to infinity, which is impossible because the corresponding P-definition would give oo, thus contradicting the hypothesis. The series has, then, an infinite number of positive terms (ui) and an infinite number of negative terms (— vi). If each of the series Wl + M2 + Ms + • • • — Vi — Vi — Vi — • • • converges, the sum converges absolutely (for we could otherwise find an arrangement r such that Dr would give oo); and our theorem is proved. Let us assume, then, that one of the series, say the w-series, is divergent. We can accordingly choose ki terms from the M-series so that ti '^Ui> Vi + I, i=l then the next ki terms of the w-series so that 5Z Ui> V2 + I, and so on. Now consider the arrangement 11ui-vi+ "22 Ui - V2+ ■•-. The sum of the first 2w terms is greater than n ; and the sum of the first (2M + i) terms is greater than « + a positive term. Hence the series diverges to oo for this arrangement, and ac- cordingly the corresponding D-definition gives it the value oo, which contradicts the hypothesis. A series may be defined to be absolutely convergent in two ways: (i) if it converges when all its terms are made positive; (2) if it converges for every arrangement of its terms. Since the concept of absolute convergence is a useful one in the theory DEFINITION OF SUM OF A DIVERGENT SERIES 75 of convergent series, it is natural to ask whether we can intro- duce, correspondingly, the notion of absolute evaluability into the theory of divergent series. The two natural definitions would be: A series is absolutely evaluable if it is evaluable (i) when all its terms are made positive, (2) for every rearrangement of its terms. Consider the first definition. If the series is eval- uable when all the terms are made positive, it must be convergent ; for otherwise it would diverge to 00 , and could not accordingly be evaluable. As to the second definition; if a series is evaluable for every arrangement of its terms, it is, by Theorem 19, ab- solutely convergent. Hence neither of the definitions of absolute evaluability is useful. § 8. TESTS FOR CEsArO-SUMMABILITY As in the case of convergence, it may happen that we wish to know not what value a given series has, but whether it has any value at all. We are accordingly led to consider tests for summability. We begin by recalling two theorems which have already been stated : Theorem : A necessary condition that the series Mi + M2 + M3 + • • • be summable (C,) is T "" * Li — = 0.* n=Qo W Theorem (3): A reducible averageable sequence with a finite number of strong limit points is Cesiro-summable of order i. This is a sufificient condition for summability (Ci). We shall now consider further sufficient conditions for summability (Ci). Theorem 20: If, in an alternating series, either (a) the terms decrease monotonically in absolute value, or (b) the terms increase monotonically in absolute value, while the sum of the first n terms is limited, then the series is summable (Ci). Let the series be Mi+Mj+WsH . and s„ = Mi+«2+- • •+Mn- In case (o) we have Sim-i > Si,„+i > s^; S2m-2 ^ s^m ^ ^i- In case (b) we have Sim-i ^ S2m+i < A ; S2m-2 >. S2m> A. Hence in either case, L 52m+i exists = h; 1^52^ exists = h. By Theorem 3, therefore, the series is summable (Ci). As examples, we may take: (0 2^1 + |-f +•••, *See p. II. 76 DEFINITION OF SUM OF A DIVERGENT SERIES 77 (») i-i + ay-iiY + iiY — , ("i) I - I.I + I. II - I. Ill + I. nil - .... Examples (i) and (ii) illustrate case (a) ; (iii) illustrates case (b). Theorem 2 1 : Lei n _ n i=l M i=l 00 //ten the series ^ Ui is summable (d) ij either (a) 5„< j„+i < /I, n> Nor (b) 5„ > 5„+i > B, n > N. For C C ^ r ^1 + 52 + • • ■ + Jn-l 1 I r o 1 ^^ - -^"-^ = «L'" ^r^i J = «f'" - ■^"-•^■ Now by (a), 5„ — 5„_i > o, and 5„ < .4. Hence Jj 5„ exists. 71=00 Similarly for case (b). 00 Theorem 22 : Le< a series 2 w; satisfy the conditions (a) //te 5erw5 is summable (Ci), (6) |5„| = |Mi + M2 + • • • + Mnl < .4, and let a set of positive constants e,- be given such that either (c) Ci > Ci+i or {d) Ci < c,+i < yl, i > N; then the series BiUi + C2W2 + • • ■ is summable (Ci). By (c), L e„ = fe, and e„ > k. n=oo 00 If Jfe =0,2^ CiWi is convergent by a well-known theorem,* and 1=1 hence is summable (Ci). If ^ +0, let S„ = e„ — ^ > 0. Then go 5b > Jb+1 > 0, and L 5„ = o. Accordingly* the series S SiUt 00 is convergent, and hence summable (Ci). But ]C *«< is sum- mable (Ci) by (o) ; so that *See Goursat-Hedrick, Mathematical Analysis, p. 349, § 166. 78 UNIVERSITY OF MISSOURI STUDIES 00 ^{5i + k)ui = 53ej«,- is summable (Ci). Similarly for case (d). If in the preceding theorem we put CO ^ Ui = 1 — i + i— I---. we obtain: Corollary l : // the terms of an alternating series monotonically decrease in absolute value, the series is summable (Ci). This is Theorem 20, case a. Corollary 2 : If the terms of an alternating series remain limited, and increase monotonically in absolute value, from some point on, then the series is summable (Ci). Since, if l5„| < 4, then \u„\ = ls„ — 5„_i| < 2A, this corollary includes Theorem 20, case b, as a special case. Before proceeding to sufficient conditions for Ces^ro-sum- mability of order higher than the first, we shall prove the follow- ing theorem,* which we shall soon need. Theorem 23: If V = vi — Vi-\- va — Vi + • • • isan alternating series whose terms decrease monotonically in absolute valtie, then the Cauchy-product of V by the series i — i + i — ! + ••• is summable {Ci). By Theorem 20, the series V is summable {C^; hence the product is, by Theorem (j), surely summable (C3). We wish to show that it is summable {Ci). {Vi - Vi -\- V3 - Vi + ■■■){i - I + I - I • • •) = Vl— {Vi + Vi) + (»1 + 1)2 + J's) - • • • . The sequence corresponding to this product series is: (a) Vu — Vi-, Vl + V3; — {Vi + Di); {vi + V3 + Vi); ■■ ■ * More generally, if U and V are two alternating series whose terms de- crease monotonically in absolute value, then the Cauchy-product of U and V is summable (C2). The proof is similar to that given for Theorem 24. DEFINITION OF SUM OF A DIVERGENT SERIES 79 and the sequence for CesJlro's first mean is: m ^:;^-: ; :•••• Let us write the odd and the even elements of this sequence: n(vi - Vi) + {n - i)ivs - Vi) + • ■ • + {v2„-i - Vi„) hn — [w(d, - 2n -V2) + {n - \)iv3-Vi) -\ + (l'2n-l - V2„)] + (Vl + V3+ ■ 1 • +f2n+l) '2n+l — _ I . 2M + I Now (vi - Vi) + {V3- Vi) + ••■ + {Vi„-i - i;2n) + • • • is con- vergent; for if s„ denotes the sum of the first n terms of this series, we have 5„_i < s„ < vi, since f„+i $ v„. Since L s„ exists, J 5l + 58 + • • • + S„ also exists, i. e., _ n(Vl — Vi) + (n - l)(tl3 -V^) + •■■ + (Z)2n- 1 - i'2n) T ^, _Lj ' ' = Ju2f2n 71=00 ^ n=a> exists. Furthermore, since L I'll exists (owing to the relation 71=00 O < Hn+i :$ Vn), n=co and hence y t^l + f 3 + • • • + t'2n+l ^ , Thus Li '211+1 "" -L* '"" ■ 2M + I n ' 2W + I' and each of these limits exists. 8o UNIVERSITY OF MISSOURI STUDIES Thus by Theorem 3 the sequence /3, having two and only two limits of equal weight, is summable (Ci). Hence the sequence (a) is summable (C2) ; which we wished to prove. If, in addition to the hypotheses of the preceding theorem, l^Vn = o, then and T Vi + V3 + • ■ ■ + ZJjn+X , J-J i = / = o, 2W + I JLt hn+l = lu hn- Thus we have the theorem, due to Hardy: Theorem m;* The Cauchy-product of a convergent alternating series whose terms decrease monotonically in absolute value to o, by I — i + i — i + --- is summable (d). We now return to sufficient conditions for summability. Theorem 24: Let mi — M2 + ws — «4 + • • • be an alternating series, «,• > o, and Aw.t > o; then (a) if A^Ui < o, the series is summable (C2) ; and (b) if in addition Ij Am„ = o, the series is summable (Ci). Case (a). Consider the series: Ui — Ami + Am2 — Ams + • • •. Since Am, > o, this is an alternating series, and since A^m,- < o, either A^m,- = Aui+i — Au, ^ o, or the terms decrease mono- tonically. Hence by Theorem 23 the Cauchy product (mi — Ami + Am2 — A«3 + • • •)(! — I + I — I ■ ■ ■) which is = Ml — (mi + Ami) + (mi + Ami + AM2) — • . • = Ml — M2 + M3 — M4 + • • • is summable (Cj). Case (6). Here the series mi — Ami + AM2 — AM3 + • • • * Bromwich, Infinite Series, p. 350, ex. 9. This is a special case of Theorem 27, below. t Aui = «,+, — ut; A'Ui = A(A»~'«,). DEFINITION OF SUM OF A DIVERGENT SERIES 8l satisfies the hypothesis of Theorem M, since the terms decrease monotonically to zero. Hence the product series ui — 112 + us — W4 + • • • is summable (Ci). Thus, for example, the series l-(i+|) + (i+i + ^) I — log 2 + log 3 - • • • are summable (Ci) ; while the series I - 2 + 3 - 4 + ••• 4^+1 2' + 1 3l±l + 23 4 are summable (Cj). Theorem 25 : If in the series mi — «2 + «3 • A*M,- > o, A*+>M.- < o, then the series is summable (.Ck+2); if, in addition, L A*M„ = o, then the series is summable (Ct+i). Let r - I + I - ••• = ^, dt = A*Mi - A*M2 + A'ms - • • • . do = «i — "2 + "3 — • ■ • • Then do = A{ui — di) di = 4 (Ami — ^2) dk-i =.4(A*-'«i -dk) Substituting the value of di in the expression for do, do = Aui — A^iAui — di). Substituting for ds, ds, and so on, in turn. Ui > o, 82 UNIVERSITY OF MISSOURI STUDIES do = Aui - A'Aui + A^Ahii - • • • ± 4*A*->Mi =f A>'dk. Now die is an alternating series whose terms decrease monoton- ically in absolute value. Hence dh is summable Ci, and A''die is summable* (Ck+i)- Since do ± A''dic consists of a finite number of terms each of which is summable (C*), or of lower order; it fol- lows that do is sammable (Ct+2) , and the first part of our theorem is proved. If we now further assume Ij A*m„ = o, n=oo it is seen that d* is convergent, and A''dk is summable Ck+i. It follows, accordingly, that do is summable Ck+i- * It can readily be proved that A'' is summable (Ct). § 9- THEOREMS ON LIMITS The object of this section is to emphasize the value, from a practical point of view, of Theorem ii, which we restate for the sake of convenience: Theorem n: // (i) Ija,(w)=o, for alii, n (2) L Haiin) = I, n=oo 1=1 (3) either ai{n) > o, or'Zlaiin)] < k* 1=1 (4) J^Sn = S, or + 00, t n=oo then n L 2 ai{n)Si = 5, n—tc i=l or + 00 respectively. We have pointed outj that many of the definitions of sum- mability are special cases of this theorem. But this theorem applies also to many other theorems on limits. To illustrate, we shall take some of the theorems from Bromwich's Theory of Infinite Series.^ Theorem n: If Bn steadily increases to 00, then L An T -"n+l -^ f I Bn 71=00 Bn+l Bn provided that the second limit exists, or is + °o. * See note (2), page 46. t See Theorem no. X See pages 43-46. § Pp- 377-388. 83 84 UNIVERSITY OF MISSOURI STUDIES To apply Theorem li,* we write: fliw = ^; a.w = — 5 . « > I- Since n Y^aiin) = I, t=i and since it follows from the hypotheses that Ij a,(M) = 0, and o, n=oD we may apply Theorem ii,* and say: If IjS„ = s or +00, then " A\ " A — A -i A L Z)oi(«)^< = ^ + L S R = L ^ = s or +00. 71=00 i=l -Ol n=QO i=l -On n=QO -^n Theorem o: If the sequences (s„), (/„) converge to the limits s, t, then -r 5i/„ + 52 + <2 + • • • + o, since by hypothesis Cn+i < c„; and in case (6), " I 2k<(«)l = "i; — [(^1 — Co) + 2(c2 - Ci) + • • • + n(c„ - c„_i) S''* + (« + l)Cn] since by hypothesis c„+i > c„; i. e., " I 2 IfliWI = "^ [- (Co + C, + • • ■ + C„_0 + (2K + I)C„1 i=0 2(m + I)C„ ^ WCn / W + l \ v-, V « / = - I + -^:^^;^^^ < 2 -^H^ ( -^-^ ] < ^. Hence in either case (a) or (6), we have: " I Ij 2fl<(M)<^< = Ij "i; [(Co — Ci) o; and in case (b), 11\ai{n)\=-^—\ -(co+ciH |-c„_i) *=» ''' + ^ (2io+26l+ ■ • • +26„_, + 6„) J = - I + 2 C„ < 2K. Now we have: {vrty^'+it-t) X [ba+bi) j^_j_ j^ + Li X^ ai{n)ai = L t=0 + ( 7^^ ~ A^ ) (6o5o + &l5lH h^n-lSn-l) + ^ (Mo + &lSlH \-bn-\Sn-l+b„S„) = L/ -S (Co^o + Ci5i + • • • + C„_i5„_i + C„S„]. 1=0 Thus in either case (a) or (6) we have the theorem established, since n L X< ai{n) 0,t Ijai„ = o, i=l n=ao and i/ L S„ = s, then L 5„ = 5. n=oo n=oo To prove this, we observe that by substituting the expression for Sj in the first expression given for S„, and equating the re- sulting coefficients of Si to the coefficients of 5i in the second expression for S„, we obtain ainOlnn + Oi n-lttn-l n + Of „_2D!„_2 „ -|- • • • -|- auUin = Sin- Adding these equations for i = i, 2, • ■ ■ «, we get: n n— 1 Otnn ZJ (^in + OCn-l n 2j Oi n-1 + ' * " t=l <=1 i=i n + «;n 2^ di,- -f- • • • -f ai„ • au = zZ bin or OLnn + Oln-l ,.+ ■••+ Oljn +•■• + Olln = I. Thus the numbers «;„ satisfy all the conditions of Theorem 11 ; and our theorem is proved. * Thus, if i4i is (Ct) and At is (Ci), then At is a generalization of Au if l> k;i.e,, if when Ai gives to (i„) a sum, then ilj will give to (f„) the same sum. t Thus {H,) and (G) are equivalent in scope; i. e., if either definition applies to Sn and gives it the sum s, then the other definition will also apply and give the sum 5. X See page 49, including footnote. DEFINITION OF SUM OF A DIVERGENT SERIES 9 1 Now assuming a„„ 4= o, and considering the formula ainOlnn + Cj „_ia„_i „ + • • • + flji «,„ = bin as n — T + I linear equations in the (n — t + i) letters au, «<+!, .. • • • ttnn; the determinant of the system of equations is fflnn O fln-1 n fln-1 n-1 Oil. fll 1.-I so that On = ann^n-l n-1 " " • All + O, I Oin = fliia>+l i+1 ■ ■ ■ <2nn ■P OiiP'i+l 1+1 • • ■ Ann Ann O fln-1 n fln-1 n-1 fli+l 1 '^•+1 n-1 din 0,i n_l bnn O 6„_i (Ji+l i+1 61+1 n fli 1+1 bin We may then restate the previous theorem as follows: Theorem 27: 7/ a,-„, &,„ are numbers satisfying conditions for A-evaluability, and D an 4= O, Uin = an ■ ■ • a„„ > o,* L a.n = o; one? if then L 5Z ain^i = ^, n=ai t=l w L 2 ^inii = 5- See p. 49, footnote. 92 UNIVERSITY OF MISSOURI STUDIES In particular, let o,-„ be the Ceskro coefficients for (C), r{r + i) • ■ ■ (r + n — i — i) _ Cr+n-i-l, n-i (n —i)\ "'"- C.+„_i,„_, ~ (r + i)(f + 2) •••(r + «-!) ' so that on evaluating the determinant D, we obtain I / r(r - i) \ "in = — I bin — rbi+i, „ H — - — bi+2, „ • • • + (— iybi+,, „ 1 , da \ I • 2 / or, using the notation r{r — i) {bin — bi+l, „)r = bin " rbi+l, „ H 77^ ^'-l-a. »•••+(" l)''^i+r, n Oin = {bin — bi+l, n)r= ~~ [(^tn" ^i+l, n)r-l— (^i+l, n — &i+2, fi) r-ll- Uii dii It is evident that Lain = o if lubin = o; n=ao 7i=oo hence we may say: Theorem 28: If 6,„, corresponding to a definition B of evalua- bility, satisfies the condition {bin — bi+\, „)r > o,* then if the sequence (5„) is siimmable (Cr), it is also evaluable according to the B-definition. If we let bin be the coefficients for summability (27 r), i- e., , ,. , {i,n)r-2 , {i + I, w) ,-2 (n, »)r-2 nbin = (t, «)r-i = ^ 1 73—^ 1- • • • ^ , t t + I n where then {i, «)i - (i+ I, w)i = 7, (i, «)p - (i + I, n)p = {i, n)p-i * The condition S l(6»n — 6i+i, «)r| < X is sufficient. 1=1 DEFINITION OF SUM OF A DIVERGENT SERIES 93 Now n{bi„ - bi+i, „)i = [{i, n)r-i - (i + l, n)r-i] = — —, r ,, , X Ji, n )r-2 ii+i,n)r-2 (,i,n)r-2 + (i,n)r-3 n{bin — bi+i, „)i = : — = rrr-: — 7 . i i+ I l(t + l) Assume /, , X Piii, n)r-2 + Piii, w)r-3 + ' " " + Pi(i, n)r-i-l n{b,. - &,,x. „), = — i(i+o..-(i+i-i) • Pi > o. Then n{bin — bi+l, „)i+l = n [{bin — 6.+I, n)j — (6i+l, n — 6i+2, n);l ^ ipx{i, n)r-2 + (pi +jpi)(i, n)r-3 + • • • + Piji, w)r-,-2 iii+i) ■■■ {i +j) _ (Tlji, n)r-i + 0-2(t, w)r-3 + ' ' ' + (Tj+lji, w)r-,-2 i{i+l)---ii+j) ffi > o. Hence by mathematical induction ,, . , Plii, n)r -2 + Piji , w)r- 3+ ••■ + pjjj, tl) r-i-\ nibin - bi+x, „),• = i(i + i)...(i+^-_i) . Pi > o, and accordingly (bin — ii+i, n); > o. Thus, by our last theorem, we may say: Theorem q: If the sequence (s„) is summable (Cr), then it is also summable (Hr).* The value of Theorem 27 is shown by its special cases, theorems 28 and Q. We shall give still another special case, Theorem p, due to Hardy, t * This theorem has been proved by Ford, Am. Journal of Math., Vol. 32, 1910, and by Schnee, Math. Annalen, Vol. 67, 1909. The converse which has been first proved by Knopp, inaugural dissertation (Berlin, 1907), can also be proved by using Theorem 29. t Quarterly Journal, Vol. 38, 1907, p. 269. Hardy states that the first part of the theorem had been given by Cauchy. See p. 87 for another proof. 94 UNIVERSITY OF MISSOURI STUDIES // a, > o, bi > o, n n An = 'Zlai, Bn = ^bi, Li 5„ = 00, J^A^ i=l i = l 71=:00 71=00 and if either at ~ fli+i 00. or and if also then Let bi . bi+i b„ fl„ -<—- and ^o, LaiSi + • • • + a„5„ ~i — i ; — z — = s, fli + • • • + a„ blSl + • • • + bnS n , b,+ ■■■ +bn = 5. and OCin = _ ai_ , _ bi_ ann • ■ • an On O an-1 fln-l O-i+l Oi+l fli a,- o 6„ O &„_i fli bi Since it follows that If further B„]_ '\ai ai+ij] Ij-B„ = 00, 71 = 00 Ij a.n = O. bi ^ 6i-K DEFINITION OF SUM OF A DIVERGENT SERIES 95 then ai„ > o. If bi ^ bj+i ai ~ a,+i' then \a„ a„_i/ a„ J = ^ [- i, - 62 • • ■ - 6„_i] + ^ ^ (^„_, + ^„) -On Dn an = — 1+2= < — 1+ 2K, Since 5- < 7C • -7- . Thus Hardy's theorem is proved* by applying Theorem 27. f Let us now return to the questions of page 89. The answer to the first question is found in Theorem 27, which is seen to give sufficient conditions that one of two definitions of summabiHty be a generalization of the other. Though these sufficient con- ditions are fairly simple, and prove useful in leading to impor- tant theorems, it would seem extremely desirable to have suf- ficient conditions that D > cj To answer the second question, we need only observe that if we can prove by Theorem 27 that definition (^4) is a generalization of definition {B) and also that definition {B) is a generalization of definition {A), then {A) and (-B) will be equivalent in scope. Now let (5„) be summable by the definition {A) and (i„) by {B), and let one definition be a generalization of the other. * The proofs for this theorem, given by Hardy (loc. cit.) and by Bromwich, Infinite Series, p. 386, are longer. t See p. 49, footnote 2. t See p. 91 and p. 49 footnote. 96 UNIVERSITY OF MISSOURI STUDIES Then the two sequences may be added term by term, and the resulting sequence will be summable by the more general of the two definitions. 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