" 26 '^C SOME THEOREMS ON THE SUMMATION OF DIVERGENT SERIES DISSERTATION SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIRE- MENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY, IN THE FACULTY OF PURE SCIENCE, COLUMBIA UNIVERSITY. BY GLENN JAMES PRINTED BY W. D. Gray, 227 West 17th Street New York City 1917 SOME THEOREMS ON THE SUMMATION OF DIVERGENT SERIES DISSERTATION SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIRE- MENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY, IN THE FACULTY OF PURE SCIENCE, COLUMBIA UNIVERSITY. BY GLENN JAMES w PRINTED BY W. D. Gray, 227 West 17th Street New York City 1917 -S3 SOME THEOREMS ON THE SUMMATION OF DIVERGENT SERIES by GLENN JAMES. 1. The Definition of the Sum of a Divergent Series. The various definitions of the sum of a divergent series are based directly or n indirectly upon the use of so called convergence factors.* A series 2 a.p is trans- formed into 1 n (1) 2 apfp (x„X2, ...,Xfc) vfhevG if, (Xi, X;,, .... xj.) is such that the "subsidiary sequence," (1), converges 00 for all values of x^, x^, . . ., x^ on certain specified ranges. The sum of 2 a, is then defined to be f P =^ ^ n (2) S= L L ... L L 2 ap fp (xi, Xg, .. ., Xfc) Xi -> Li Xj -^ Lj Xfc — » Lt n -> CO p = 1 For brevity we will write the above as follows : n (3) S= L L 2, apfp (x<), (i= 1,2, 3, ...,k) x< — > Li n — » 00 1 For example the series can be summed by taking, 1 fp (xi) = , X > 0, c> 1 f.px Then n (— l)p*^ 1 S= L L 2 = L = X— >oon— >ool c"* X— >ooc*-|-l We might, however, choose * C. N. Moore: Trans. Am. Math. Soc. (1901), p. 300. S. Chapman: Quarterly Journal (1912), V. 43, p. 1. L. L. Smail: Dissertation, "Some Generalizations on the Theory of Summable Series," Columbia University (1913), Chap. II, p. 4. t We may have Xj.=n as in Cesaro's mean, § 3. In this case there would be only k limiting processes. 371JJ0 3 . 1 fp(Xi)= , X>0, C>1 qP/M In this case S = 3^. Evidently the usefulness of the general definition depends upon the restric- tions which are placed upon fp(xi). (a) Consistency in the general theory of series demands that any method of summing divergent series should be such as to give the ordinary sum when applied to convergent series. This is called the "condition of consistency." * (It is evident that the first method in the above example does not satisfy this condi- 1 tion, since L = 0. But the second method does since, X — ^ 00 c"' 1 11 L = 1 ; L = and is always positive.) f X -^ 00 C^'''^ p — > 00 cP'' c^'* (b) Evidently the sum of any series should be the function whose expan- sion gives the series, when such a function exists uniquely.^ (c) When we are summing a series of variable terms, which converges uniformly, it is desirable that the "subsidiary sequence" should converge uniformly with respect to the variable of the original series. If such be the case, a series is said to be "uniformly § summable" by the given method. In general it seems desirable to seek methods of summing divergent series which make the sum bear the closest possible analogy to the ordinary sum of convergent series. However, in this paper we shall be concerned only with the above restrictions, which are indeed most essential. In the next article a mean for summing divergent series is developed which satisfies the condition of consistency and sums uniformly convergent series uni- formly. This mean includes the most interesting form of Euler's Transforma- tion for slowly convergent series ^ as a special case, and with a slight modification becomes a generalization of the means of Frohenins and Holder. Properly divergent series have been shown to be non-summable by the well known methods of Borel, LeRoy, Cesaro or Holder. || The third article of this paper extends the notion of proper divergence to a class of oscillating series which are non-summable by the above methods and, in fact, by the above general definition of p. 1, when certain restrictions are placed upon it. These restrictions * Bromwich: "Theory of Infinite Series," p. 269. t See article 3, Theorem VIII. tPringsheim: Encyklopedie, Bd. I, A. 3, 39. § Any series is said to be "uniformly summable" when the subsidiary sequence con- verges uniformly; see Hardy, Transactions Cambridge Philos. Soc, Vol. 19 (1904), p. 301. ^ Bromwich : Loc. cit., p. 55. (l L. L. Smail : Dissertation, Columbia University (1913), Chap. III. are shown to make the definition satisfy the condition of consistency and to sum uniformly convergent series uniformly. 2. A Mean for Summing Series. Consider the series (4) ai + ag + aa + . . . 4- ar = Sr where L Sr = s r — > 00 The plan followed in the first part of the succeeding development is as fol- lows: Take the means m at a time, of the terms of (4) beginning with the first, then the second, and so on; make corrections to maintain the identity; repeat the process on the new series n-1 successive times and take the limit as r increases indefinitely. For n 5 1 and r 5 m,* we have, (5) ai+a2+a3+...+a„ a^+^z+^i-h • -- +^m^x ^3+^4+^5+- - •+^m*2 m m -f ...+ l+ar_m+2+ar.n,+3+ar (m— 1 )ar+ (m— 2)ar.i4- • • ■+^r.m^2 m m (m — 1 )a,4- (m — 2)a2+ . . . +am-i — Sr- m The sum of the first m terms of (5) is, (ai + 2a2 + 3a3 + ... -f- m a„, -f- ... -f- agm.J/m From the expansion, by actual multiplication, of (1 +x + x2-f ... -f x»"-^)' we see that the above numerator is just this expansion with x' replaced by a^+i, q = 0, . . ., (2m — 2). Hence we will denote (ai + 2a2 + 3a3 + . . . + ma^ H f- a-zm-^) by '(a, + ... +a«) Taking the means of (5) we now obtain, with this notation: (6) + +--- + m* m^ m^ * Although this and the following identities hold for m=:l, this is a trivial case. Hence it will be understood throughout this paper that m>l. (m — l)ar+(m— 2)ar_i+. . .+ar.m+2 + + m (m— 1 ) (ar+ . . . 4-ar_m^i) + (m— 2) (ar.i+ . . . +ar.m) + (ar_m,2+ • • • +ar-om,3 ) = Sr (m— l)ai+(m— 2)a2+. . .+a,„.i m (m— 1) (ai+. . .+a„.) + (m— 2) (a^-\-.. .+am.i) + - • . + (am-i4-- • -+^2^-^^ Evidently the sum, as before, of the first m terms of (6), r being chosen suf- ficiently large, will give a form in a^, a^- • -, ^3m.2 which is just (1+x-j-. . .^-x^'O^ with x« replaced by a^+i, (q=0, 1, 2 ... 3m — 3). Analogous to the procedure above we denote this form by ^{a..^-}-. . .+am). Similarly the nth mean will give us forms "(ai+. . .+am), which are equivalent to (l+x4-. • •+x'""^)" with x^^rag+i, q:=0, 1, ..., n(m — 1) since there are n(m — l) + i terms in the nth power expansion of the sum of m terms. We can now write out the identity resulting from the n — 1th repetition of the above process. It is as follows : (7) "(ai+--.+am) "(a2+...+am,i) m" + ...+ y^r-n(m-i)~T' • ' • ~T~ar_n(m_i)+in_i ) "r -K r m" = Sr (m— l)ai-f(m— 2)a2+- -.+3^-1 + m (m— 1 ) (a, + . . . H-a„,) + (m— 2) (aa-f . . . +a^, J + . . . + (a,„_i+ . . . +a„„., ) + ...+ (m— l)«-^(a,+ ...+a„) + (m— 2)"~M;a,+ ...+a„„i)^ m' . . . -]-" {a.m-1-f- . . . -f-agm-o ) where, (m — l)ar+(m — 2)ar_i+. . .+ar_m+2 (^ — l)(ar+. . .+ar_m+i)4- RV=: + m m^ (m 2)(ar_i+.. .+ar.m) + . . .+(ar_m+2+-- •H-ar-am+s) + ...+ ■(m— 1) "-Har+- • .+ar.„,.J + (m— 2) »-i(a..,+ . . .+a,._„.) + . • ~r"' (ar_m+2~r' • •~rar_2f»+3) and r^n (m — 1) + 1. We will now seek the limiting form of (7), as r increases. Consider first the remainder R',-. Since the series (4) converges, for every c, there exists an r' such that for r 5 r' _ £5 la,._,!, [iz=0, 1, 2, ..., (n— l)(m— l)+m— 2] whence for r > r' RV5 (m— 1 ) + (m— 2) + ... +1 (m— 1 )m+ (m— 2)m+ . . . +m + (m— l)m"-i-|-(m— 2)m»-i+. . .+m"-^ m» n(m— 1) and it follows that (8) L R'r=0, for every n and m. r— >oo The terms «(ai+. . .H-a„,) «(a2+. . .+a„,,i) «(ar_n(m-i) + . • .+ar.(„.i)(m-i)), m" m" when added term by term take the form (9) "[(ai+- • •+ar-«(m-i)) + (a2+. . . +ar_„(m_x).i) + . . • + (am-f- • •+ar_(n-i)(m-i))]l/m". This can be written, (10) "[Sr_n(»i-i)+ (S;_n(m-i)+i Si)-)-- • • + (S,_(n_i) (m-i) Sm_i)]l/ni'* The last term in the expansion of ( 10) would be Sr_n ( m-i ) +n (m_i) +i Sn(m-i)^^Sr+i Sn (m_i ) CO Now since 2 a„ = s 1 (11^ •L' Sr-mm-i) — •L' Sr_(>H_i)+i — ... — L. Sj- — L Sr+j^ — S, r— »oo r— ^00 r— ^00 r-^oo m and n being fixed. Whence (12) L "[Sr-«(m-x) + (Sr-»(m_i)+i— Si) + . . .4-(Sr.(«.i)(m-i)— Sw-i)]l/m» r— »oo ="[s+(s— sj + . . . + (8— s„,.,)]l/m» ="(Rc,+Ri+...+R„..,)l/m» where Ro=s and Ri is the remainder after the ith term. From (8), (12) and the fact that L Sr=s, we see that the limiting form of r— >oo (7) is (m— l)ai+...4-a„,.j (13) «(R<,+ ...4-Rm_01/ni"=s + L 1^1 "-Hm—l)*(a,+ ...+a„..,) + (m— 2) *(a2+..+a.„) + ..+*(a»<.,4-- ••+».»,-,) 2 . i.i m**i or (14) "(R„+. . .+R;„.,)l/m"r=.s— „,S'„ where mS'» denotes the second term of the right member of (13). Theorem I. When the series aj+a2+. . .+a«+. • • converges to s, n— >oo To prove the theorem, it is sufficient to show that the left member of (14) converges to zero. Expand "(R„+. . .+R,„_i)l/m'», denoting the coefficients by Co, c^, . . ., c„(,„-i) and write the expansion as follows : Ro+CiRi-f- . . . +c/,-Rfc Cfc^iRfr^iH- . . . +R„(m.i) (15) + , m" m** noting that Co=c„(„,.i) = l. Now for every e there exists a (k) such that (16) lR)i-.x|^c/2, i=l,2,3,... 00 since 2a„:=s. 1 Whence (17) |CA-,,R..,+ .-.+R«k 8 Consider the first k-f-1 terms of (15). Since L l/m"= L Ci/m"=...= L ct/m»=o n— > 00 n-^ 00 n— > oo there exists an N such that (18) lR<,+c,R,+ .?.4-CfcRfcl ge/2, n>N. m" Combining (17) and (18) we have (19) l''(Ro+. . .+Rm-i)l/ni"l ^e, n>N', N' being equal to or greater than k either or N. Thus the theorem is proved. m— 1 From (12) and (14). (20) s— ;„S'„="[s+(s— s,) + . . . + (s— s..a)]l/m" ="(s+s+...+s)l/m»— »(0+s,+ ...+s,„.i)l/m'» =s— "(0+s,+ . . . H-s^.i) l/m» where ,„S„rr:(0+Si+. . .+Sm_i)l/m". Then (21) t«S„=:mS'„ or written out in full n(n±l)* (22) (n SiH s,+ . . . +Sn(m.i) ) 1/m" 2! = [(m— l)a,+ (m— 2)a,+ ...+a„_Jl/m+ n— 1 r (m— 1) *(ai+...+a^) + *(a2+...+a„,^i) + ...+<(a;„.i + ...+a2„.2) i=l m* Theorem II. When the series 3^4-^2+^3+- • • converges to s n-^oo m— »oo m-^00 m,n— >oo m,n— »oo That L ,„S„=s follows from (21) and Theorem I.* n— >oo Consider L ,„S„= L mS'n—s. The proof of Theorem I suffices for m— » 00 m— > 00 these cases provided we fix n and let m vary, replace the conditions n>N and n>N' by m>M and m>M' and note in connection with (18) that the first k The plus sign occurs for m>.2. coefficients in the expansion of (1+x-f . . .+x'"-^)'» never change with m provided m>K. Finally consider L ,„S„: m,n— »oo L mS'„=s. Since n and m both increase, m,n-^co replace, in Theorem I, the conditions n>N ancf m>]Vr by the pairs n>N, m>M and n>N',m>M', respectively, N' and M' having been chosen so that, (m — l)n 5 k and the inequality (18) holds for n>N' and m>M'. This proof then suffices for the above cases. Theorem III. When a series of variable terms converges uniformly it is uniformly summable by mSn and mS',, with either n or m or both increasing. The proofs of Theorems I and II suffice for this theorem when (16) is replaced by (Rt+i) ^ e/2, i=l, 2, 3, .... for all values of the independent variables on the intervals considered. This, of course, is just the definition of uniform con- vergence. Thus it has been proven that the condition of consistency is satisfied and uniformly convergent series are uniformly summable by ,„S„ and „,S'„ with m fixed and n increasing or with n fixed and m increasing or with both m and n increas- ing in any manner whatever, m and n being on the range, (1, 2, 3 . . . ). Upon the basis of „,Sn we define the two following sets of means : (A) n(n±l)s2 n Si-| f- . . . -}-S„(m_i) S'^' S'^' ,S':? 2! n(n±:l) 2! n(n±:l) l/m"=„,S„ 1/m" Hmo 1 2! S['--i]4_ I Q [r-D l/m" where m is fixed and n increases. (B) n(n±l) nsi-l- 2! S2"T" • • • ~\~Si7um-i) }S,r ':^s,.= • n(nzt:l) n 1 «Jw n 2 '-'" n^ • • • 1 // ()„.i)j „ n 1 ^)i 2! n(n±l) l/m"=rt,S„ l/m" (r-UC 4_ I [;• I] C l/m** . 2! where n is fixed and m increases. Theorem IV. The means (A) and (B) satisfy the condition of consistency 10 for all values of the parameters m and r in (A) and n and r in (B). Consider first the means (A). We have proved (Theorem I) that the above theorem is true when r=:l. Hence when (23) Sj, Sj, ... Sn ... converges to s so does (24) „,s'i\ „s^ ,„s':,' ... Applying Theorem I to ,nS^^\ we see that the sequence cr2] C[2] c[2) converges to s whenever (24) does, hence whenever (23) does. The theorem follows by induction. The argument for (B) is the analogue of this since the theorem is proven foi r=:l in Theorem II. Corollary. If (A) or (B) sums a series for any value of r. it sums th< series to the same sum for every greater value of r. Theorem V. The means (A) and (B) sum uniformly convergent series uniformly. This theorem has been proven (Theorem III) for r=l. The proof for any i is the same as the proof of Theorem IV, except that the convergence of eacl sequence is uniform. The means (A) and (B) can not sum properly divergent series. This becomes evident for rr=l when we observe that for every k the sum o the last n(m — 1) + 1 — k coefficients of (l+x+. . .+x"'-^)" divided by m", the sum of all of them, has the limit unity. The arguments for „,S»f''' and ^''^mS,, ar( the same as that of Theorem IV except that we have proper divergence of eacl sequence instead of convergence. If n and m increase together, in any manner whatever, in (A) and (B) Theorem IV, its corollary, Theorem V and the above remark on proper diver gence hold for these methods. This follows from Theorem IT and Theorem II by arguments similar to those used for (A) and (B). Euler's Transformation for Slowly Convergent Series, with x=l,* is a specia instance of „,S'„ namely gS',,, hence is identically equal to .S',)\ (=2S„). Fo putting m=2 in (22) gives (25) n(n — 1) 1 aj a^+a- a-,-f-2a.,+a., ,S„=(n s,+ s,+ . . .+s„)— =— + +— ^-- • • + 2! 2'- 2 2^ 2' (n— l)(n— 2) aj+(n— l)a,H a.,+ . . .+a„ 2! * L. D. Ames. Annals of Mathematics, Series 2, Vol. 3, p. 188. Equation (1) of thi article should read s=:Uj+u,+U3+. . . "11 The right member, which is Euler's form, is thus expressed in terms of s^, s^, S3 . . A Mean Having Frohenius' and Holder's Means as Special Cases If we add to m S1+S2+ . . . +Sm-i Si+S.+ . . . +S„ ,Si, { = , we have m m which is Frohenius' mean.f (When mS^ or Frohenius' mean has a limit, the other one has the same limit, for mSj: Si4-S2+ • . . +Sm-i m — 1 m— 1 m In general "(a,+ ...+a«) if we add to mS» and form a set similar to (B) upon the basis m" of this sum we have a generalization of both Frohenius' and Holder's t means, n(n±l) iOn n s, 2! S2 I • • • ~rS)i(m_i) Add n(n±l) ai+n a,H aj-j- . . . +a, 2! •(«n_i)+i I 1/m" 1/m" and denote the sum by m(S)i». Then (26) „(S)„= n(n±l) Sj-f-n SjH ^3~r ' ' • "T"Sn(»i-i)+i 2! 1/m" Analogous to (B) we formulate; (C) m(S)„='«[s,+ . . .+s«] l/m»=«(S), c^3(S)„=»['lHS)«+...+',V(S)„]l/m* M(S)„=»[-''l"(S)«+-..4-^^^KS)„]l/m» where n is constant and m increases. Theorem VI. The means (C) satisfy the condition of consistency and sum uniformly convergent series uniformly. tBromwich: Loc. cit., Article 51, Ex. 2. JBromwich: Loc. cit., Article 123. 12 From (21) m(S)„=„,S',,+"(a,-f . . .4-am)l/m'»=m(S')n, say, Then we can write S m \^ ) «^^^ S til ( o j n or whence or "(S+...+S) I (,►5^11 — S Hi^o )n m" (27) "[(s— s,) + ... + (s— s„,)ll/m"=s— „,(S')„ "tRi4-. • •+R».]l/m«=s— «(S')„ Now, upon the basis of (27) results for «,(S')„ and m(S)„ analogous to Theorem I, the first part of Theorem II and Theorem III are proven just as these were. Theorems for (C) analogous to Theorems IV and V then follow. When n=l, (C) becomes, ',!,HS)x=[s,+s,+ ...+s,„]l/m l^HS),= [^l'(S),+'^>(S):+...+'''(S)Jl/m ',:;'(S),= ['-'(S),+ '-'(S),+ ... + '«"(S)Jl/m which is exactly Holder's mean. Holder, however, used T'i' where we are using 'i'(S)^. When n=r^l, in (C), we have Frohenius' mean. There is an interesting relation between 2(S)n and Borel's original definition,* consequently between the latter and Euler's Transformation. Horel's original definition is n x» (28) L L ve-s,,^^ — X— >oo n^oo o n ! (29) .,(S)„ Introduce the factors 1, (29) and replace 1/2" by n(n-l) [Sj+n s,H S3+ . . . +s„, J X x^ n n^ 1 2.! X' n» 1/2" as multipliers of the successive terms of (1+-)" n We then have,t ♦Bromwich: Loc. cit.. Art. 114. t This suggests a generalization of Borel's original definition upon the basis of (C), but the resulting forms are exceedingly complex. 13 n(n-l) (30) I Ls,+n/n s^x-\ S3 xV2 !+ . . . +s„^i x'Vn"] (1+-))" Denote this form by s (x,n). Then, < L L s(x,n)= L [s,-fs,+S3xV2!+...]e-'^ x-^00 n— »oo x— »oo n = L L 2 e-'^Sn+i xVn ! x^oo n— >oo o while L s(x,n)=r: L 2(S)n x=n— >oo n— »oo Thus the sum by Borel's form and that by 2(S)n, when they exist, can be derived from the same function, s(x,n), the difference being in the manner of taking the hmits. Comparing (29) with (25) we see that a^ a^+a, ai+2a2+a3 "-^(ai+ao) "(a^+a,) ,(S)„=-+ + +. . .+ + 2 2- 2^ 2" 2" "(ai+aj That is, o(S),, is Euler's Transformation. Whenever the latter 2« <"-^'(a,+a2) "(a.+a^) converges to a limit, converges to zero, hence does. Con- 2" 2"^^ "(a.+aj sequentlv converges to zero. Hence when Euler's Transformation 2" gives a sum, it is the same as L s(x,n). x=n— >oo The means, (A), sum series for which (B) fails just as Euler's method sums certain series for which Holder's and Cesaro's methods fail. However, it is desirable to have different methods to use on different types of series provided they do not give contradictory results.* Illustrations of the Use of (A) and (B) on Some Familiar Series * See (b) p. 4. 14 (1) The series l4-x+x--|-x^ . . . which is summable by (B) or Cesaro's or Holder's methods only for 1>X5 — 1 is summable by (A) for xx> (1 — 2'"+^), where r is arbitrary. That is, the sum o n— »oo 1 — X l-!-x4-x-+. . . is for every value of xK* For this binomial form results from taking the nth mean of 0, a^— (2a)^ . . ., (— l)"^^(n a)^ (see pp. 6-7). ar=l, K=l, gives the series (2') 1— 3-f5— 7-f...=0 a=:l, K=:2 gives the series (2") 1— 5-fl3— 25-j-...=0 (3) The series 1—1-1-1— 1-f sums to 1/2 by (A) using any m or by (B) using any n, or by either with m and n both increasing. Here Si=:l, So^O, 83=1, ... .S'!'^ n+0+...-f (l-\-( l)»(»i-i)+i l/m"=„,Sn When m is even the sum of the even coefficients in (l-|-x-|-. . .-j-x'""^)" equals the sum of the odd, and when m is odd, the sum of the even equals the sum of the odd minus unity. In either case if O and E denote the sum of the odd and of the even coefficients, respectively, E+0=r2E- l_|_(_l)m.l That is, m«=2E- l_|-(_l)m.l * A. M. Kenyon: "Some Properties of Binomial Coefficients," Proceedings of the Indiana Academy of Science, 1914, p. 2. 16 Whence m" E: l4-(— l)«-i 2 4 Substituting this value of E in the above form of S, 1 2 l + (— l)*"^^ m" 1/m" Hence (4) The series n— >oo m— >oo »»,n— »oo 2 a— (a+d) + (a+2d)— (a+3d) + . . . 2a— d stuns to ,On — s,=:a, s.r=d, s^=a+d. s^^ — 2d, S5=a4-2d, . . . n(n— 1) n(n— l)(n— 2) n(n— 1) (n— 2) (n— 3) n a ■ — d-\ (a+d) ^2d 2! 3! n...(n— 4) 4! ■(a+2d)-... 1 2» 1 rn(n— 1) n(n— l)(n— 2) n(n— 1) (n— 2) (n— 3) _9" a) I — — : 3! n. . . (n — 4. a— 2"— d-^ 1 -2 2 12! 3! 4! -2.. 5! 1 2- SimpHfy the coefficient of a. In the coefficient of d add and subtract Then collect one-half of the sum of the even binomial coefficients. This gives, 2 d 2 2» n ln(n— l)(n— 2) -+ 2 2 3! 11 d +- 2 2" — 1 2 n(n— 1) n+ 2 2 2! 3n(n— l)(n— 2) 4n...(n— 3) 5n...(n— 4) n + +... + (_!)«_ 2 3! 2 4! 2 5! 2 17 a d 2» 2 2» 2' 2 n+l n(n— 1) n(n— l)(n— 2) o — n+2 3- 2! 3! n...(n— 3) 4! J The bracket is zero, n 5 2, since it results from taking the nth mean two at a time of the terms o, 1, — 2, 3, — 4, 5, . . ., ( — l)"n, (see pp. 4-6). Whence, 2a— d 2^H— — — — ■L< 2^'> 4 n— >oo (4') ar=l, d=l gives 1—2+3— 4+ 5... = 14 (4" ) a= 1 , i\-^2) gives l-4+7-10+...=->4 It is interesting to notice that the sum of a— (a+d) + (a+2d)— (a+3d) + . . . < is positive, zero or negative according as d = 2a. (See (2'), p. 16.) > This result could have been obtained by using (B). 3. Non-summable Series.* Consider the series (1) a,+a.,-fa3+...+a„+... and the sequence n (2) Sj, s... S3, . . . , s„, ... where s„=r:2a„ 1 Definition. Denote by N., the class of series which are such that for every c and every M there exists an m 5 M such that for every n 5 1, n 2 (s;„,p— c)5 0, (orgO). P=l The class N.. contains all properly divergent series and such series f as * Presented to the American Mathematical Society, April 29, 1916. fThe sequences of S's are respectively: 1, 0. 2, 0, 3, 0, 4. 0, ... and 2, —7. 4. — 2, 6, — 3, .... In either case for a given C we can choose m such that S,„^j+S„,^^>C, n consequently 2 (S„,^„ — C)50, n^l. p=l 18 1-1+2-2+3-3+... + (-1)' 2n+l+(— l)«*i 2—3+5—6+8—9+ . . . + (—1 ) "^'- 6n+l + (— l)"-i The method of procedure in this article is to place sufficient restrictions upon the general definition, (1), § 1, to make the members of N« nonsummable.J, then prove upon the basis of this general method, that the specific methods mentioned in § 1 can not sum series in the class N.,. It is. incidentally, shown that the general definition thus restricted satisfies the condition of consistency and sums uniformly convergent series uniformly. Definition (1) § 1 can be expressed in terms of s^, s^, ... s„. ... as follows: (3) S„=fia, + f2a.+ . . .+f»_ia„_,+f„a„. (3') (3") (4) where whence (5) -fiSr+l.(S,— Sj) + . . .+f„_i(s„_i— S„_,)+f„(S„— S„..j) ^Si(f 1— f2)+s,(f — f,) + ...+s„.,(f„.,— f„)+sj„ :Sj*i+S2, p=l,2, 3, ...,n— 1, S= L L n— 1 2 Sp«^p(Xi)+S„fn 1 i=l,2, 3, ... k Now let (6) 0p=s,a — c, (see definition, p. 18), and replace s,, by c+0p in n— 1 S„= 2 Spp+s„f„ 1 Then (7) n — 1 n — 1 S„=c 2 <^„+ 2 0p^,+s,.f„ 1 1 But n— 1 n— 1 2 $,= 2 (f„_,-f,) = f,-f,. 1 1 t These restrictions are chosen so as to include in the restricted general method those special methods by which the class of series N^ are non-summable, these specific results having been obtained independently. 19 Whence n— 1 S„=cfi+ 2 0p*p+(s„— c)f„ 1 or m n — 1 (8) S„=cf,+2®p o for n>m+l>M. m+1 m+1 (See definition, p. 18.) Then in the Hght of (4) we can write n— 1 (10) 2 ®„^p=s'm^i(^m^i *»i+2)+s'».+2(*m+2 ^»U3) + --'- m+1 + S'„_2 ($„_2 *„_i ) +S'„_i*„.i n— 1 and 2 ®p% will be positive or zero provided (^m+p — ^m+p^i), m+1 (p=:l, 2. . . , (n — m — 2) and ©„.i, are positive or zero. This will be the case if we assume that 20 (U) ^m^p, [=f„,^p(Xi) — f,».p+i(xj)], P^O, never increases with p, (for any set of values of Xi, i=l, 2, 3, ... k), M having been properly chosen, and that (12) L f„(xi)=o, for all values of x^. n— »oo For from (12) L 4>„=o and from this fact and (11), n^oo (13) *„_,= (f„.,— f„)5 0, n5M+l Finally consider the last term of (8), namely 0„f„. Evidently (14) f„(xi)5 0, n^M+l. Hence 0„f„ can not remain negative, for 0„ can not remain negative and n increase without limit in absolute value since 2 @m+p 5 o, n 5 1. P=l We have thus shown that S,, either oscillates or can be made greater than any positive quantity C, by a proper choice of m. n The argument for series such that S (s„,^p — c) ^ o, is the same as the above. P=l Theorem VII. Every series of the class N« is non-summable by the defini- tion n S= L L 2 a„f,(xO, i=1.2. . . . k, Xi— >Lt n— >oo p=l where I. L ip{xi) = \, for every p Xi-^Li II. L f„(xi)=o n— >oo and III. For every set of values of x^, x^, . . ., xt, there exists an M such that the sequence, ( Im+p I»i+p+ij) P^— ^'J; A? ^> • • • never increases. Theorem VIII. Definition (1) § 1 under restrictions I, II and III satisfies the condition of consistency. Let a,-|-a2+ . . . +an4- • • • converge. 21 Since, s — Ri=Si, s — Ra^Sg, ..., s — R„=:s„ where the form of s„ in (3") (p. 19), can be written, after collecting the coefficients of s, n— 1 Su^sfi 2 Rp(fp— fp^.l) — Rnf» and for any m, (^n — 1), we can write S„=s f,— 2R,(fp— f,,J- 1 By I of Theorem VII, L x«— >Li n— 1 2 Rp(f,— f,,J+R«fn m+1 sf— 2Rp(fp— f,,,) 1 = s The last term on the right converges to zero for "n— 1 5 Rp(fp— fp,,)+R,.f„ m+1 <€ Im+i Im+2"T~ tni+2 • • •~il>i provided m is chosen large enough to make III, Theorem VII, hold and also to make |Rp|m. Now from II and III, there exists an m such that * o^fpm whence there exists an m such that n— 1 2 Rp(fp fp+i)+Rnf»i m-f-1 <€, n>m+l. Corollary. The general definition restricted by I, II and III sums uniformly convergent series uniformly. Theorem IX. Le Roy's method can not sum series in the class Ng. Le Roy's f definition is (15) n r(pt+l) : L L 2 ap, t— >1 n— >co p=o r(p-|-l) l>t>o. ♦Choose an m such that |fm+pl P>o- Then the sequence, f„,, f,„^^ never in- creases and converges to zero according to II and III. Hence f„,^p>o. t Annates de Toulouse, Ser. 2, Vol. 2 (1900), pp. 323-327. 22 Comparing with the general definition of Theorem VII we see that r[(p-l)t+l] fp(Xi, X2, . . ., Xfc) = fp(t)= r(p) Condition I of the above theorem is satisfied since . r[(p-l)t+i] L =1 t-.l r(p)) r[(n-i)t+i] Condition II holds for L =0*, l>t>0. n^oo r(n) In order to show that Condition III holds it must be shown that for every t, l>t>0, there exists a P such that for every p>P, r(pt+l) r(pt+l+t) r(pt+l+t) r(pt+l+2t) (16) % •r(p+i) r(p+2) r(p+2) r(p+3) Since p is a positive integer, the denominators are, respectively, p!, (p+1) ! and (p+2) ! By the identity r(x+l)=xr(x), r(pt+l+t) r=(p+l)tr(pt+t), and r(pt+l+2t)=(p+2)tr(pt+2t) whence (16) reduces to t r(pt+l)— tr(pt+t) 5 tr(pt+t) r(pt+2t) P+l or r(pt+l) t5t r(pt+t) 1 — t- r(pt+2t) 1 (p+l)r(pt+t)J r(pt+2t) r(pt+l+t) Since the right hand side is less than t it will be sufficient to show that r(pt+l) r(pt+t) 5 2t ♦Smail: Dissertation, Columbia University, 1913, pp. 38-39. 23 By Sterlings formula,* r(pt+l) e-^pf^i-''Upt-\-t—l)p*^t-^/^V2ll (17) L — =1 p-^oo e-p'(pt)p'^i/-V2n r(pt+t) Or simplifying, r(pt+l) t— 1 (18) L e^-'(lH )^'-^/-(pt+t— l)'-^=rl p-^oo r(pt+t) pt But and t— 1 L (1-^ yui/2^^t~i p-^oo pt L (pt+t — l)'-^=:o, since t 00 " p ! If Borel's original definition sums a series, his integral definition gives the same sum,J and if Borel's integral definition sums a series, LeRoy's method gives the same sum. § By Theorem IX. LeRoy's method can not sum series of the class N,. hence neither of Borel's methods can.^ *Bromwich: Loc. cit., page 462. t See note, page 28. JBromwich: Loc. cit.. Article 114. § Bromwich : Loc. cit., Article 116. ^ It can be shown by an argument similar to that of Theorem IX that Borel's methods are special instances of the general method, see note. p. 26. 24 Theorem XI. Cesaro's mean * 5= L n-^oo A^";' r(r+l) r(r+l)...(r+n— 1) where S^r,^ =s„+r s„_.^-\ S„.2+ • • • H So 2! n! and A':;' = (r4-l)(r+2)...(r4-n)/n!, can not sum series of the type Nj. First consider r 5 1. Sn can be written n n(n — 1) n la^ ( 19) a„+ a,+ a,+ . . . -f r+n (r+n— l)(r+n) ' (r+1) (r+2) . . . (r+n) tivei )n c Then.t It is convenient, now, to let p run from o to n, instead of 1 to n, in the gen- eral definition of Theorem VII. This complies with the usage in Cesaro's mean. n(n— l)...(n— p+1) (20) fp(xO = f,(n)= . P^n (r+n— p+1 ) (r+n— p+2) . . . (r+n) Now fp(n) may be written, n — i n — p+1 r+n — p+1 (r+n — p+2) r+n Each of these fractions approaches unity as n increases, and there are exactly p of them. Hence, (21) L fp(n)=rl, for every p.$ n— >oo that is. condition I of Theorem VII holds. Again, h(n): (r+1) (r+2). (r+n) r+1 r+2 r+n— 1 n+r *Bullitin des Sciences Math. (2), V. 14 (1890), p. 119. t In this case x^.^n. See note, p. 1. ifln this mean fo(n)r=:l. 25 1 Every fraction on the right is equal to or less than unity and has the n+r limit zero. Hence (22) L f„(n)=o, n-^oo that is, condition II is satisfied. Condition III. From (19) it is easily seen that ip — (p^^, p=0, 1. 2, . . ., is, * r rn rn(n — 1) r+n (r+n— l)(r+n) (r+n— 2) (r+n— 1) (r+n) These terms never increase when r 5 1. Finally it is necessary to note that ip — f,,+, is always positive since (13), page 21, does not, in this case, follow from conditions II and III. It would, also, suffice to prove L f„_i(n) =0. n— >oo If Cesaro's mean sums a series with any value of r, it gives the same sum with any greater r.* Hence Cesaro's mean can sum series in N., for no value of r.f Theorem XII. Holder's mean can not sum series of the class Ns. It has been shown by Knopp $ that Holder's mean 1 sv; V'J= — -So+s,+ . . .+s„)=- n+l A';; 1 Vf}= (T*'' +T7 + . . . +T T) n+1 1 'r;>= — (x<'-j'>+T"-:"+. . .+T"'-'0 n+l gives a sum only when Cesaro's gives the same sum. Thus the above theorem is a consequence of Theorem XI. The means § (A), (B) and (C) of article 2 do not satisfy condition III of Theorem VII for the polynomial coefficients first increase and then decrease and the maximum coefficient moves out the sequence s^, Sj, ... as n increases. Hence we cannot choose an m such that fm+p — fm+p+i never increases as p increases. * Chapman, Proc.'s, London Math'l Soc, Vol 9 (1911), pp. 369-409. fWe have as a corollary of Theorems IX, X and XI, that LeRoy's, Borel's (see note, p. 24) and Cesaro's methods satisfy the condition of consistency and sum uni- formly convergent series uniformly but these are familiar results. :|:K. Knopp, Inaugural Dissertation (Berlin, 1907), p. 19. § The question of the non-summability of series in N, by these means will be con- sidered in a later paper. 26 BIBLIOGRAPHY Ford, "Studies on Divergent Series and Summability," Michigan Science Series, Vol. II. This treatise contains an extensive bibhography to which I will only add, L. L. Smail's Dissertation, "Some Generalizations in the Theory of Divergent Series," Columbia University, 1913. 27 VITA Glenn James was born October 2nd, 1882 ; attended Vincennes University 1901-1903; received the A. B. degree from Indiana University 1905 and the A. M. 1910; studied at Chicago University two summers, and at Columbia one. summer and throughout the year 1915-1916; instructor at the Michigan Agricul- tural College 1905-1908 and at Purdue University since then; member of The American Mathematical Society and the Indiana Academy of Science ; has pub- lished "The Accuracy of Interpolation in a Five Place Table of Logarithms of Sines" in The American Mathematical Monthly, October. 1913, in conjunction with Professor A. M. Kenyon, and "A Note on the Sum of the Remainders of a Convergent Series," American Mathematical Monthly, November, 1916. He wishes to express his gratitude to Professor W. B. Fite for the inspiration received from him and his very careful criticism of this dissertation. 28 14 DAY USE RETURN TO DESK FROM WHICH BORROWED LOAN DEPT. 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