C-NRLF *B 530 185 SOME GENERAT.IZATTONS IN THE THEORY OP SUMMABLE DIVEEGENT SERIES BT LLOYD LEROY SMAIL , DISSERTATION Submitted in Pakti^ll Fulfilment of the Requirements for the Degree OF DocjTOR OF Philosophy, in the Faculty of Pure Science, Columbia University PRESS OF THE NEW ERA PRINTINQ COMPANV LANCASTER, PA. 1913 GIFT OF SOME GENEKALIZATIONS IN THE THEORY OF SUMMABLE DIVERGENT SERIES / BT LLOYD LEROY\SMAIL DISSERTATION Submitted in Partial Fulfilment op the Requirements fob the Degree OF Doctor of Philosophy, in the Faculty of Pure Science, Columbia University PRESS OF THE NEW ERA PRtNTINO COMPANY LANCASTER, Pa. 1913 INTRODUCTION. In this paper, a process is given which leads to four general methods of summation of divergent series, and each of these methods includes as special cases several of the known methods. These latter are sufficiently indicated with their connections in the course of the discussion. It is shown that, in accordance with these general methods, every convergent series is sum- mable and the generalized sum is equal to the ordinary sum; whilst a properiy divergent series is not summable by these methods. Uniform summability, and the continuity of uniformly summable series and their term by term integration and differentiation, are discussed. Of the general theorems ob- tained, applications are made to the particular methods of Cesaro, Riesz, BoREL, LeRoy, and the so-called Cesaro-Riesz methods of Hardy and Chap- man. The methods of proof employed throughout are simpler than those hitherto used. In this way the essential properties of the various known methods are brought out, and greater uniformity of treatment is secured. S s, lU 478370 SOME GENERALIZATIONS IN THE THEORY OF SUMMABLE DIVERGENT SERIES CHAPTER I. The Problem of Divergent Series. 1. Let us first seek what meaning can be assigned to an infinite series. In the first place we are to understand by an infinite series a symbol such as 00 (1) ao + ai + 02 + • • • + a„ + • • • = 2 «n . To assign a meaning to this symbol, the simplest and most natural way is to form the expression n and when lim Sn exists, to take this value, the so-called sum of the series, as n — ^00 a substitute for the series wherever the series occurs in calculations. Thus the method of convergent series is simply a particular method of associating a definite number with the series, and using this number in place of the series in calculations. This limit, lim *„, however, only exists for certain series, while there are many series, the so-called divergent series, for which this limit does not exist. In order to be able to use such series, we must then find some method by which we can associate a definite number with the series, so that we can use this number in place of the series in calculations. Our problem of divergent series is then to associate with such a series a number, which we call the Sum* of the series, which should be defined in such a way that the resulting laws of calculation agree as far as possible with those of convergent series. If the series has variable terms, we wish to associate with it, not a number, but a function, which shall satisfy the above condition. Any definite method by which we can associate with a given series a Sum is called a method of summation. 2. Chapman t has stated a "general principle of summability " for any v * For convenience, we write the sum of a convergent series, in the usual sense, with small 8 , while this associated number here referred to, we write Sum, with capital S. \y t Quarterly Joum. of Maih., Vol. 43, p. 4. 1 / -, J « J/ 3 ♦,.. '_ ■ 2 LLOYD L. SMAIL: SOME GENERALIZATIONS IN THE infinite form as follows: "When the sequence of finite forms, which defines or generates an infinite form, does not tend to a limit as the variables tend to infinity in the assigned order through the sequences of values constituting the domains of these variables, then we may agree that the number represented by the given infinite form is to be the limit of a sequence of associated finite forms, different from the members of the original sequence; the second sequence must of course be judiciously chosen, so that the limit to which it tends is usefully related to the original sequence. The number of its variables may be the same or greater than the original number; and the additional variables, if any, may or may not be required to tend to infinity." Applying this to infinite series, we shall form sequences involving the terms of the given series, but which have limits even when the sequence Sq, Si, *2 , ' ' ' f Sn, • ' • has no limit. We shall not only form subsidiary sequences depending upon one index, but also sequences depending upon two indices (double sequences), and have simple and double limits of such sequences. An infinite series will then be said to be summahle by any particular method if the corresponding subsidiary sequence has a limit. If the terms of the given series are functions of a variable u , then this series will be said to be uniformly summahle with respect to u in an interval ( a , 6 ) , if the subsidiary sequence converges to its limit uniformly with respect to u in that interval. 3. Methods of summation have been classified by Chapman* into yarametric and non-parametric. A method of summation is parametric if the associated sequence used to determine the Sum contains a parameter of some kind upon the values of which depend whether or not the series is summable. If the associated sequence contains no such parameter, the method is called non- parametric. 4. As already explained, the Sum given by a method of summation is to be used whenever the given series is such that Sn has no limit. Now «„ has a limit for convergent series (in which case Sn has a finite limit) and for the properly divergent series (where «„ has the limit + » or — « ) ; our methods of summation are then designed primarily for the oscillating series. But con- vergent series and oscillating series may occur in the same piece of work, and so our methods of summation should be such that they will apply to convergent series and give a Sum equal to the ordinary sum. Likewise, when the methods of summation are applied to a properly divergent series, they should give a Sum equal to + oo or — <» , according as *» has the limit + oo or — oo . We formulate these requirements in the following conditions, which we call the conditions of consistency: (1) When the given series is convergent, the Sum given by any method of summation must exist and coincide with the sum. Moreover, if the method * Quarterly Journ. of Math., Vol. 43, p. 8. THEORY OF SXJMMABLE DIVERGENT SERIES. 3 is parametric, the Sum must exist and coincide with the sum for all values of the parameter for which the method is applicable; (2) When the given series is properly divergent, the Sum given by any method of summation must be + oo or — oo , according as lim *„ is + «> or — 00 ; (3) When the given series is uniformly convergent, it must be uniformly summable; and this must be so for every value of the parameter if the method is parametric. CHAPTER II. Some General Methods of Summation. 5. Having given a divergent series ^a„ , we start with the expression u [n] (2) Zo./,, where n is always taken positive, and [ n ] denotes the greatest integer ^ n , and/» is a function of certain variables and parameters to be specified presently. We shall study the limiting properties of this expression, and find out under what circumstances these limits will satisfy the conditions of consistency. We first enunciate some of the cases which may occur and which we shall study: I. fv may be a function of n and also of a parameter k; then, keeping k fixed, we seek the simple limit [n] (3) lim ^Ovfvin, k) . n-^oo ti=0 II. /, may be a function of a variable x; we then seek the repeated double limits n (4) lim lim 2 Or /» (a;),* n (5) lim lim 52 o» /t> ( a; ) , or the Pringsheim double limit n (6) lim Sa„/v(ar), n, X— >so 11=0 where n , x tend to oo independently but simultaneously. III. /„ may be a function of two variables n and p; we then seek the repeated double limits (7) lim lim 51 Or /• ( n , p ) , n > flo p — >oo »=o (8) lim lim J^a^fv (n, p), p — >« n^->ao ti=0 * The case x -> oo may be regarded as a general case, since any other, x—*a, can be reduced to this by transformation. 4 THEORY OF 8UMMABLE DIVERGENT SERIES. 5 or we may seek the Pringsheim double limit • fn] (9) lim 52 «v/* (w» V)y n, p— >« t>=0 or we may seek the double limit along a path F (10) lim Va„/„ (n, p), where n , p tend to <» simultaneously though not independently, but in such a way that a functional relation F(n,p) = holds between them.* In place of starting with the expression (2), we may start with the expression (11) jls^f., »=0 « where 5p = JZ <*» • Then we have the case i=0 IV. fv may be a function of x , and we seek the repeated double limits (12) lim lim i:*„/. (x), n — >« X > oo r=0 n (13) lim Wm ^ Sv fv {x) . Let us now examine each of these cases more in detail. Case I. 6. Let fv {n, k) he defined for all positive integral values of v , including , for all positive values of n , and for a certain range of real values of the param- eter k . Suppose further that/» {n, k) satisfies the following conditions: 1° 0<_fv(n,k)^l (or every V , n , k; 2** when n and k are fixed, the sequence fOf flf fz) ' ' ' }fv , ' ' ' is monotonic decreasing; 3** lim/p (n, k) = I for d fixed; n > X 4" foin,k) = l; 5' lim /[nj(n, k) = 0. * See Hardy and Chapman, Quarterly Joum. of Math., Vol. 42, p. 187. 6 LLOTD L. 8MAIL: SOME GENERALIZATIONS IN THE When limit (3) exists and has the value S: (30 \\mX.a,fAn,k) = S, 00 then we shall say that the series ^Ovis summable (/) with Sum S . We now proceed to show that this method of summation satisfies the con- ditions of consistency. It may be noted that this is a parametric method. oe 7. Theorem 1. If the series 2J a* w convergent with sum s , then (3) (»i lim ^Ovfv (n, A:) n—^tn r=0 exists and is equal to s for every value of k for which fv is defined; so that every convergent series is summable (I) with Sum equal to sum, and part (1) of the conditions of consistency is satisfied. Put 1 -fv (n, k) = gr, (n, k) , then by condition 1°, we have 0^ gv (n, k) ^ 1, by 2° the sequence go, gi, gz, " - , gv, • • • is monotonic, and by 3°, lim gv (n, k) = . n — >oo Put Gin,k) =2lav9v{n,k) = 2lav - Zlovfv (n,k). v=0 0=0 r=0 Then we have to prove that (a) limG(n,Jk) = 0. « — >ao We may write (^) G(n,k) = ^a„gr,-{- Jl (h Qv Let us consider first the second sigma of (6). By Abel's lemma,* [»] 2 OvOv {n,k) < Ag < A, where A is the upper bound of 23 Ov * See Bromwich, Theory of Infinite Series, § 23. THEORT OF SUMMABLE DIVERGENT SERIES. for r^ N+l, N + 2, .-., [n], and g is the upper bound of the Qv for v= N+1, •••, [n]; by condition 1°, ^ ^ 1 . Now since S Op is convergent, we can choose No so that A < € / 2 (or N > Nq, where e is any arbitrarily small positive number. Hence (c) v=N+l (hgv foT N> No,n> N + 1. Now consider the first sigma of (6). By condition 3°, when N is once fixed, for every given positive number e', we can find an integer rriv such that for g„{n,k)<€' foru= 0, 1, 2, ..-, iV. Let m be the greatest of the finite set of integers viq, mi, • • • , m^; then for n> m, each gv {n, k) < e' {v = 0, -- - , N) . Choose then e: »=0 ^Ovgv (w, k) y If ^12\av\ • gv(n,k) < €'J2\av for n > m . e'< i)=0 2 El a. zlcLvgv € <2 From (c) and (d), we now see that \G{n,k)\ < e where N' is > NqOt m. Hence f or n > m . for n > N', w [n] lim z2 (hfv {n, k) = lim E «» = « . n-»ao ti=0 n-^oo i>=0 The above argument has not explicitly involved the value of ^ , so that the theorem is true for every k for which /„ satisfies the conditions of § 6 . It may be remarked that all the conditions of § 6 are not required for this proof; we need only conditions 1°, 3", and 2° can be replaced by the require- ment that the sequence (/» ) be monotonic. 00 8. Theorem 2. If the series z2 (h (u) is uniformly convergent with respect [n] to u in the closed interval (a, b) , with sum s{u) , then ^av{u)fv{n, k) tends 8 LLOYD L. SMAIL: SOME GENERALIZATIONS IN THE to its limit s (u) uniformly with respect to u in the interval (a, b); so that c/ uniformly convergent series is also uniformly summable (I), and part (3) of the conditions of consistency is satisfied. Using the notation of § 7, we must prove that G (n, k, w) -» uniformly in ( a , b) . We find, as before, 2 <^ »=JV+1 (w) • <7„ (n, k) < A{u) • g Nq and for every m in (a, 6) . We then have (c) 22 Ov (u) gv {n, k) No and for every w in ( a , 6 ) As before, we find So* (w) • gv (n, k) I e=0 < e'Slot, (w) f or n > m , Let K be the upper bound of f or w in ( a , 6 ) ; take then id) S I Ot. ( w ) €< S Ot, (w) • ^t. (n, fc) e: ti=0 m, and for every m in ( a , 6 ) . Hence G(n,A;,w)— »0 uniformly with respect to u , and our theorem follows. 9. Theorem 3. If the series ^a^ is properly divergent, so that lim 5n = + « , then [n] lim X a„/, ( n , k) = + oo ; N-^ae vssO hence a properly divergent series is not summable (I) with finite Sum, and part (2) of the conditions of consistency is satisfied. THEORY OF 8UMMABLE DIVERGENT SERIES. 9 We have Z^ttvfv = Sofo + (Si — So)fl + (S2 — Si) fi + • " + (*[„] — 5[„]_i)/[„] v=.0 = 5o (/o — /l ) + 5l (/l — /2 ) + H *[n]-l (/[n]-l " /[n] )+*[n]/[n] • [n] [nl-1 (14) . ■ . S at>/t) = S *» (/t. — fv+l ) + «[n] /fn] . »=0 f=0 Since lim 5n = + 00 , if ii is any arbitrarily large number, we can find m such that we can put Sv = K -\- ry ioT V > m, where r„ > 0. Put Then (14) becomes [n] m [n]-l ^avfv{n, k) = '^Svhv{n, k) -jr K "^ hv{n,k) v=0 v=0 t)=m+l [ w]— I m + S ryh„{n,k)+Sin]fin]in,k) = ^{Sv — K)hv{n,k) [nJ-1 [n]_l H-X S ^t>(n, k) + S r„^„ (n,.^) + 5[n]/rn] (w, A;). But Z ^v(n,^)= (/o-/a)+(/i-/2)+ .-. +(/[n]-l-/r„]) =/o -/[«]. [i»3 m (a) .'. ^avfvin, k) = J2 (Sv — K) hv(n, k) + K {fo{n, k) — f^n] {n,k)] + S r„hv {n,k) -\- Sin]f[n](n,k). By condition 2°, hv (n, k) is ^ 0; by 3°, lim A^ (n, k) = 0; n— >oo byl°, /[n] (n, A:) > 0. Using these results, and conditions 4", 5°, we get [n] [n]-l lim 2^ Ovfy (n, k) = K -\- lim J^ r« A^ ( w , ^ ) + lim *[„] /[„] ( n , A; ) . > X. Since K is arbitrarily large, our theorem follows at once. 10 LLOYD L. SMAIL: SOME GENERALIZATIONS IN THE For this proof, all the conditions l°-5° of § 6 are required. OB 10. Theorem 4. If the series ^a^ {u) is uniformly summahle (I) with respect to u in an interval {a, b) , with Sum S (u) , and if the terms a„ ( w ) are continuous functions of u in {a, b) , then S (u) is a continuous function of u in (o, 6) . Since [n1 00 S (u) = lim S«» {u)fv (n, k) = S^v (m)/v {n, k) , n-^w r=0 it follows that S ( w ) is the sum of a uniformly convergent series of continuous functions, and hence, by the properties of uniformly convergent series, S {u) is continuous. The theorem may be proved directly as follows:* Put [n] 22 a„ {u)fv {n, k) = Sn(u) . t>=0 We have \S{u+h)-S{u)\S\S{u-\-h)-Sn{ui-h)\-\-\Sn{u-\-h) -Sn(u)\'\-\Sniu)-S{u)\. Now since lim/S„(w) = S{u) n->oo and the approach is uniform with respect to w in (o, &) , we can find N such that f or n > iV , and \S(U) -Sn{u)\ <|, Siu-^h)-SAu + h)\<~, where e is any arbitrarily small positive number. Again, since Sn {u) is a sum of a finite number of continuous functions, we can find d such that \Sn{u-\-h)-Sn{u)\<^ for U|<5. Hence we obtain \S{u+ h)- S{u)\ < € for I A| < 5, which shows that S {u) is continuous. ao 11. Theorem 5. If the series 2lciv{u) is uniformly summable (I) with respect to u in the closed interval {a, 6 ) , with Sum S (u) , and if {ci, c^) is * See Chapman, Quarterly Journ. of Math., Vol. 43, p. 12. But THEOKY OF SUMMABLE DIVERGENT SERIES. 11 an interval contained in (a,b) , and if the terms av (u) are integrahle in ( Ci , C2 ) , then the series obtained by integrating the given series term by term over the range S{u) du. We can write where 8n (u) is such that we can find N so that for n > N , I 5„ ( W ) I < € , however small e may be, for every uin {a,b) and [n] Sn (w) = ^ttv {u)fv (n, k). /»cj /»ci /»rj .'. I S (u) du = I Sniu)du-\- I 8n (u) du VCl *J C\ VCl = ^fv{n,k)- I ttv (u) du -\- j 8n{u)du. I 5n(w)rfw^ I I 5n (W) I • I N)y .'. lim I 5„ (w) dw = 0, lim2/p(7i,A;) I a,,(w)oo t)=0 t/cj «/c] Hence the series J^ I av {u)du\?, summable (I) with Sum I S{u)du* «/ci t/ci Observe that this proof holds for any function f^ (n, k) , such that Sn (u) approaches a limit S {u) uniformly, even though it should fail to satisfy all the conditions l^-b" of § 6. 12. Theorem 6. // the series ^av{u) is summable (I) with Sum S (u) , X then if the series zJ <^» ( w ) obtained by differentiating term by term the given series is uniformly summable (I) with respect to u in an interval ( a , 6 ) , with Sum «— ?oo 0=0 ti=0 THEORY OF SUMMABLE DIVERGENT SERIES. 13 and n n lim lim ^Ovfv (x) = lim ^ a» . OB Hence, limit (4) can only exist when the series ^ a„ is convergent, and will not give a method of summation of non-convergent series. We need then consider limit (4) no further. From the well-known theorem:* " If lim Up, q exists and if lim ttp, q exists for every q, then p, q-^co J>-»ao lim (limOp, g) = lim ap, q," q-^ao p^oo p,q— ^00 we see that if lim Op, q exists for every q , then lim ap, q can only exist when lim lim ap, , exists. n Then since lim ^ a„ /^ ( a; ) exists for every n , it follows that the Pringsheim double limit n (6) lim 2I«»/» (x) n, x—^ao v=0 can only exist when the repeated double limit n (4) lim lim zlcivfvix) exists, i. e., only when the series S av is convergent. Hence the limit (6) will not apply to non-convergent series, and we need not consider it further. 15. When the limit (5) exists and is equal to S , n (5') lim lim ]Ca»/» (a;) = S, z— >aa n— >ao t»=0 we shall say that the series 2 a^ is summahle {II) with Sum gp* '" »s monotonic, and that lim g^ (x) = for V fixed. Put then since we must prove that ••0 n n G{n,x) = Y,av - Hovfv (ar), lim lim G (n, «) = 0. We shall first show that the Pringsheim double limit lim G(n,x) = 0. Writing (? ( n , X ) in the form (?(n,x) = So,^, (x) + S a»^»(a;), we first apply Abel's lemma to the second sigma, and obtain E fl»y«(«) "^ ^ ' g "^ ^ for every X, where A is the upper bound of for r == iV + 1 , • • • , n , and i^ is the upper bound of the terms g, for c = iV + 1 , • • • , n , so that g^\. Since 2 a» is convergent, we can choose N^ so that, for every «, for iV > iVo, we have A < t/2; hence (fl) for N > No and for every x . 2 a»y. («) X^ , we have g, {x) < t' for v = , 1 , " • , N . Let X be the greatest of the finite set of integers Xv, (v = , 1 , • • • , N) , then for X > Z , we have each g^ (x) < e' {v = , ' ■ • , N) . Har,gvix) ^J2\av\ ' gv(x)< e' -^lov r=0 fora;> X. If we take we get (b) €< 2Ek 2a„flr„ {x) <\ From (a) and (6), we now obtain \G{n, x)\< t so that f or a; > X . forn> iVoH- 1, x> Z, lim G (n, x) = n, at— >oo We now apply the theorem (already quoted in § 14) : *// lim ttp, q exists and lim Op, , exists for every q, then p, ?->« p-»« lim lim ap, , = lim Up, g." 5— >oo i>— >w p, 7— >« Making use of the theorem:* A convergent series remains convergent if its terms are each multiplied by factors which form a bounded monotonic sequence, we see that the series Z^ a^ g^^ ( x ) is convergent and lim G{n,x) exists. Hence by the theorem just quoted, we have and lim lim G {n, x) = , lim lim 2^ «» /» ( a; ) = lim 2l(h> = s. Not all the conditions of § 13 were used in this proof; we need only 1", 3", and in place of 2° we need only require the sequence (/« ) to be monotonic. 30 17. Theorem 8. If the series ^ av is properly divergent, so that lim 5n = + 00 , n->QO • See Bromwich, Theory of Infinite Series, § 19. 16 LLOYD L. SMAIL: SOME GENERALIZATIONS IN THE then n Hm Hm 5Zoc/p (x) = + 00 ; that is, no properly divergent serjj^ can he summable (II) loith finite Sum, and hence part (2) of the conditions of consistency is satisfied. Starting with equation (14) of §9, and proceeding exactly as in §9, we arrive at the equation corresponding to (a) : (a) Za.fv(x) = Z{s.-K)Kix) + K{fo{x)-fnix)] «=0 «=0 n-1 + H Tr, h (X) -{- Snfn {x) , where K{x) =fv(x) -/h.i(«). We find from the conditions of § 13, that lim hv (x) = 0, lim fo (x) = 1 , lim lim fn (x) = . n n— 1 .*. lim lim ^a^fv (x) = K -\- lim lim ^r„ hv (x) + lim lim Snfn (x) . But since r„ > , A» ( a; ) > , and *« > , /« ( a: ) ^ , we have n-l lim lim ^rvhg (x) > 0, lim lim Snfn (x) > , n .*. lim lim 2Zor/» (x) > i^. z— >oo n->oo tt=U Since K can be taken as large as we please, our theorem follows at once. 18. Before proceeding to show that this method of summation satisfies part (3) of the conditions of consistency, it will be necessary to make the notion of uniform summability (II) more precise; for this notion involves uniform approach to a repeated double limit. A definition of uniform approach to a Pringsheim double limit is easy to formulate. If Cp, , (w) is a function of u, and if for each value of u in an interval ( a , b) , the Pringsheim double limit lim Op, 5 (w) exists and is equal to a ( w ) , then we shall say that flp, « ( w ) , approaches its Pringsheim double limit a (u) uniformly with respect to u in the interval ( a, 6 ), if for any positive «, two integers P, Q can be found such that for every p^P,q^Q, THEORY OF SUMMABLE DIVERGENT SERIES. \a (U) — ttp, q{u)\ < € 17 for every value of w in ( a , 6 ) . 19. Theorem 9. If cip, q (u) approaches its Pringsheim double limit lim Up, q (u) = a (u) uniformly with respect to u in an interval (a, 6) , and if for each q, ap, q (u) approaches a simple limit lim ap, q(u) = ag (u) uniformly with respect to u in {a, b) , then aq{u) approaches the simple limit lim aq{u) = a{u) uniformly with respect to u in (a , b) . By hypothesis, we can find numbers P , Q such that (a) I a (u) — ap, q (u) I < toT p> P , q > Q, and for every w in ( a , 6 ) ; also we can find a number Pq (depending in general upon q ) such that (6) lap. 5 (w) - ttq (u) I <2 for p> P'q and for every u in {a, b) . Adding (a) and (6), we get I a (w) — ttq (u) I < € foT q> Q and for every u in ( a , b) , which gives the result stated in the theorem. 20. The theorem just proved suggests a definition of uniform approach to a repeated double limit, which will be useful in our later discussion of uniform summability. If for each value of u in the interval ( a , b) , the repeated double limit lim lim ap, q (u) exists and is equal to a (u) , then we shall say that flp. « (w) approaches its repeated double limit a (u) uniformly with respect to w in ( a , 6 ), if ap, q (u) approaches its simple limit lim ap, q(u) = aq (u) 1§ LLOYD L. SMAIL: SOME GENERALIZATIONS IN THE uniformly with respect to w in (a, b) , and if a, (w) approaches its limit lim ttg (u) = a (u) 3^00 uniformly with respect to w in ( a , 6 ) . n Our definition of uniform summability (II) will then be: If 2Za» (w)/„ (a;) t;=0 approaches its repeated double limit n lim lim ^a„{u)fv{x) = S {u) I— ^00 n— >so v=0 OB uniformly with respect to u in an interval ( a , 6 ) , then the series 5Z «» ( w ) is said to be uniformly summable (II) with respect to w in {a, b) . 21. We are now in a position to discuss the uniform summability (II) of a uniformly convergent series; but before entering upon that, we shall first prove an auxiliary theorem, in order not to interrupt the argument later. Theorem 10. A uniformly convergent series^ av (u) remains uniformly convergent if its terms are each multiplied by factors gr„ , provided that the sequence (gv) is monoionic, and that \ gv\ <, a constant c* Since the sequence {gv) is monotonic and \gv\ < c , g,, must approach a limit, call it g . Put bv = g — gv when ( gr„ ) is an increasing sequence, and K = gv — g when (gv) is decreasing. Then the sequence (bv) is monotonic decreasing and approaches the limit . Since Ov (w) • gv = av (u) ' g — av (u) • 6„ or av (w) - g + a„ (u) • bv, oe we need only prove thatZ]a» (u) - bv is uniformly convergent. If A {u) is the upper bound of r for r = m+ 1> •••, m + p, we have by Abel's lemma m+p '^aviu)-bv < A{u) ■ bm^i< Aiu) ■ bo. But since JIcLv (u) is uniformly convergent, we can find M such that for m> M , we have A{u) M and for every positive integer p , and for every u in (a, b) . 00 Hence the series ^Ov (u) b^is uniformly convergent in (a, b) . oe 22. Theorem 11. If the series ^aviu) is uniformly convergent with respect n to u in the interval (a, b) , with sum s (u) , then 23a„(w) •/,, (x) tends to its repeated double limit s{u) uniformly with respect to u in (a, b); that is, a uni- formly convergent series is also uniformly summable (II), and so part (3) of the conditions of consistency is satisfied. n We shall first show that X)«d(w) ' fv(x) approaches its Pringsheim double v-O limit s (u) uniformly with respect to w in ( a , 6 ) . Using the notation of § 16, we have to prove that numbers N , X can be found such that for every n> N , x > X , I G (n, X , u) \ < € for every u in {a, b) . As in § 16, we find by Abel's lemma 2 av (u) gv (x) ^ A(u) • g S A{u), for every x and for every u , where A (u) is the upper bound of X)a»(w)| for r=iV+l,---,n, and g is the upper bound of the terms gv ior v = N -\- 1 , • • • , n . Since zl^v (u) is uniformly convergent, we can find iVo such that for N > No, we have ^(w) No, for every x , and for every w in ( a , 6 ) . As before, (§ 16), \^av (u) gv {x)\< e'Yl\civ(u)\ for every x> X . I »=0 I «=0 20 LLOYD L. smail: some generalizations in the Let K be the upper bound of S | a„ ( w ) | for w in ( o , 6 ) ; and take e' < ^. Then (6) Jlav(u)gv{x) »=0 € <2 for a; > Z and for every w in (a, b) . Hence n I J^civ {u)g^{x)\< e »=o I (oT n> N > No, x> X , and for every u in ( a , b); that is, G (n, x, u) approaches its Pringsheim double limit uniformly. Qvix) evidently satisfies the conditions of Theorem 10, so that, since Sa»(w) is uniformly convergent, ^Uv (u) - g^, (x) is also uniformly convergent; that is, the limit n lim Ylciv (u) gv {x) is approached uniformly with respect to w in ( a , 6 ) . Now applying Theorem 9, we see that the limit n lim lim Z^a„(w)^„(a;) X — >ao n— ^00 v=0 is approached uniformly with respect to u in ( a , 6 ) . Our theorem then follows. 23. Theorem 12. // the series 2J«» (w) is uniformly summable (II) with respect to u in an interval ( a , 6 ) , with Sum S (u) , and if the terms a„ ( w ) are continuous functions of u in {a , b) , then S (u) is a continuous function of u in {a,b) . Put and then Z)a« (w)/« (x) = Sn. X (w), limSn. X (w) = Sx(u), n->ge lim5x(w) = S{u). We have \S{u+h)-Siu)\^\Siu-\-h)-SAu + h)\ + \SAu + h) (a) - Sn. X (W + A) I + I Sn, X (W + A) - Sn. x (w) | + I -Sn. X (W) - -Sx (W) I + ! Sx (W) -S{U)\. (d) \Sn,Au)-SAu)\ <^, THEORY OF SUMMABLE DIVERGENT SERIES. 21 From the definition of uniform summability (II), it follows that, for any given € , we can find X such that for x > Z , we have (b) \SAu)-Siu)\ <^, and (c) \S{u + h)-SAu+h)\<^^, and that we can find N such that f or n > N , (d) and (e) \SAu-hh)- Sn,x{u-[-h)\<^. Since Sn, x (u) is a. sum of a finite number of continuous functions, we can find 5 such that for | A | < 5 , we have Combining (6)-(/) , we get \S{u-{- h) - S{u)\< e for|A|<5, from which the theorem follows. 24. Theorem 13. If ^Ov (u) is uniformly summable (II) with respect to u in an interval ( a , 6 ) , with Sum S (u) , and if its terms a„ (u) are integrable, then the series obtained by integrating the given series term by term with respect to S(u)du. Using the notation of § 23, since lim5„. x{u) = Sx{u) n->oo and the approach is uniform with respect to w in ( a , 6 ) , and since 00 limS„, X (w) = Hov (w)/» (x), n— >ao it follows that the series Sx{u) = J^a„ (u)f„ (x) is uniformly convergent in ( o , 6 ) . Hence I Sx{u) du = ^fr, (x) I aviu)du. *Jex *J ci 22 LLOYD L. 8MAIL: SOME GENERALIZATIONS IN THE /. lim Hm Jlfv{x) I a^{u)du= lim S/e(a;) I at,(w)(iw= lim I Sx{u)du. Then, to prove our theorem, we must show that lim I (Sx(w)dw= I S{u)du. But We can write where i?* ( w ) approaches its limit , as a; — > oo , uniformly with respect to ti in ( a , 6 ) ; we can then find X such that for ar > X , I '/o! ( w") I < € for every m in ( a , 6 ) . /»<•» /*"» /»«t I S (u) du = j Sx{u)du+ j rfx (u) du, I T/x ( W ) fZw ^ I I r/x ( W ) I • I Z . 7ix{u) du = and lim I Sx{u) du = I S (u) du. «0 25. Theorem 14. If 23 a„ (w) i* summable (II) m n. Suppose that/» (n, p) satisfies the following conditions: 1° 0^/e(n,p)^l for every D, n, p; 2° when n , p are fixed, the sequence /o , /i , ••-,/«, • • • is decreasing; 3" lim/„ (n, p) = 1 , and when N has been fixed, we can choose no so that THEORY OF SUMMABLE DIVERGENT SERIES. 23 fv {n, p) -*l uniformly for V = , 1 , - ■ • , N , n "k no] 4* lim lim /„ ( n , p ) = 1 f or « fixed ; p— ^00 n— >» 5°- lim fv (n, p) = 1 for « fixed, for certain paths F; F 6® lim lim /[«] (n,p) = 0; 7° lim /[„] {n, p) = for certain paths F. n,p— >« F Hardy and Chapman* have shown that the limits (7) and (9) can only exist when the given series 2 a„ is convergent, so that they will not give rise to methods of summation of non-convergent series. We need not consider them further. When the limit (8) exists and is equal to S : [n] (8') lim lim Yj civ fv (n, p) = S , p—^con-^aov=0 we shall say that the series 2 a^ is summable (III A) with Sum S . When the limit (10) exists, with the value S: (10') lim ^ttvfv (n, p) = S , n, p— >oo v=0 F we shall say that the series S a„ is summable (III B) with Sum S . We shall now show that both these methods satisfy the conditions of con- sistency. X 27. Theorem 15. If the series ^a^ is convergent with sum s , then it is sum- mable (III A) and also summable (III B) with Sum equal to s in both cases, so that part (I) of the conditions of consistency is satisfied for both methods. Hardy and Chapman have proved f that when 2 a„ is convergent with sum s , then lim zlovfv (n, p) = s. It follows at once that the Hmit (10) taken along any path F will be equal to s . 00 Since /» ( n , p ) ^ 1 , the series ^ Ovfv (n, p) is convergent, so that X/ Ovfv ( n , in § 14 , we see that the limit (8) exists and is equal to s lim z2 dvfv (n, p) exists. Then from the theorem on double limits quoted n— ^00 r=0 * Quarterly Joum. of Math., Vol. 42 (1911), p. 202. t Quarterly Joum. of Math., Vol. 42 (1911), p. 202. 24 LLOYD L. SMAIL: SOME GENERALIZATIONS IN THE 28. Theorem 16. If the series X a^ (m) is uniformly convergent, with sum 8 {u) , in an interval (a, 6) , then it is also uniformly summable (III A) and (III B) with respect to u in {a, b) , and part (3) of the conditions of consistency is satisfied for both methods. [n] We shall first prove that ^a„(u)fv{n,p) approaches its Pringsheim double c=0 limit s (u) uniformly with respect to w in ( a , 6 ) . If we put 1 —fv {n, p) = Qv {n, p), and [nl G(n,p,u) = ^av(u) g„{n,p), we must prove that numbers N , P can be found such that for n> N and p> P , we have \G {n, p,u)\ < € for every w in ( a , 6 ) . Just as in § 22, we have 2-j Ov v=y+i {u)gv {n,p) <: A (u) ' g < A (m), for every p and u , where A (u) and g are defined as before. Since X! «r ( w ) u is uniformly convergent, we can find A^o such that A (u) < e / 2, and there- fore (a) N+l {u)g„ {n,p) No, for every p , and for every w in ( a , 6 ) . Now when N is fixed, by condition 3° we can choose no, po such that \ gv {n , p) \ < e' ioT V = 0, 1, " • , N , n > no , p > Pol hence Jl a„ (u) g,, {n , p) <€'2]|a,(w)| (n>no,p>po). Let K be the upper bound of J^ | a„ ( w ) | for w in ( a , 6 ) , and take e' < e / 2K . Then (6) A E t=0 Z)ar (u) gv {n, p) <2 for n > no, p> po, and for every w in ( a , 6 ) . (a) and (b) then give (c) \G{n,p,u)\ < € f or n > N' , p> Po, and for every w in ( o , 6 ) , where N' is the greater of No and no- THEORY OF SUMMABLE DIVERGENT SERIES. 25 It follows at once from the result just proved that £ a, ( w ) /„ ( n , p) ap- proaches its double limit along a path F: [n] lim ^ av (u) f„ (n , p) = s (u) F uniformly with respect to m in ( a , 6 ) ; so that the part of our theorem which relates to the method (III B ) is proved. Using Theorem 10, we can readily show, as in § 22, that as n — > oo , M zL (iv {u) fv {n , p) approaches a limit uniformly with respect to w in (a, h) . t.=0 Then by Theorem 9, it follows from the result obtained in the first part of this section, that X^a„(w)/«(n,p) approaches its repeated double limit 11=0 [n] lim lim Xa„(w)/„(n,p) = s {u) p— >ao n — ^00 r=0 uniformly with respect to w in ( a , 6 ) ; so that the part of our theorem which refers to the method (III A) is proved. 00 29. Theorem 17. If the series zlctv is properly divergent, with lim 5„= + oo , n->oo then it is not summable (III A) or (III B) with finite Sum; hence part (2) of the conditions of consistency is satisfied. Equation (a) of § 9 becomes in this case: [n] m (a) J2a„f^{n, p) = Y< (Sy - K) h {n , p) -\- K {fo {n , p) — f^] {n , p) } •=0 »=0 + S r,>hv{n,p) + sin]f[n]{n,p), where h (n, p) = fy {n, p) — /„+ 1 ( n , p) . From conditions 4°, 6°, we have lim lim ^ (n, p) = 0, lim lim/o (n, p) = 1, lim lim/[n] (n, p) = 0. p->oon— >« p— >aon->oo |}— >oo n — >« Hence W [n]-l (6) lim lim 2_/a»/» (w, p) = jfiT + lim lim ^ ryh„{n,p) p— >« n— >w r=0 p— ^00 n— >oo v=m+l + lim lim5[n]/in] in,p). p-^oo n— ^00 But since r„ > 0, K (n, p) > (by 2"), 5[„] > 0, /[n] (n, p) > 0, the last 2Q LLOYD L. smail: some generalizations in the two terms on the right in (6) are positive, so that lira lim Z^avfv(n,p)>K. p— ^00 n— >w «=0 Hence S a, is not summable (III A) with finite Sum. Again, from conditions 5°, 7", we have lim Ap (n, p) = 0, \im fo(n,p) = 1, lim /[„] (n, p) = 0. Also n, D-^ao n, p— >oo n, o— >oo F F F lim zl rr,K{n,'p)>0, lim «[„]/[«] (n, p) > 0; z' r hence 2^ a lim 22^r,fv (n, p) > /ii. n,p-^oo „—Q F and therefore S a» is not summable (III B) with finite Sum. 30. Theorem 18. If ^ av (u) is uniformly summable (III A) or (III B) with respect to u in an interval {a, b) with Sum S (u) , and if the terms o^ ( m ) are continuous functions of u in (a, b) , then S (u) is continuous in ( a , b) . The proof of the part of the theorem relating to the method (III A) is precisely similar to that of Theorem 12. To prove the other part, put- Zlav{u)fv{u,p) = Sn,p{u), then Sn, p (u) approaches its limit lim Sn, p (u) = S (u) n,p-^m F uniformly with respect to w in ( a , 6 ) , so that (O) |S(«)-Sn,p(w)|<^ for every w in ( a , 6 ) , if n , p are so chosen (satisfying the relation F(n, p) = 0) that the corresponding point ( n , p ) is sufl5ciently far along the path F . Writing \Siu+h)- S(u)\^\S{u+h)-Sn,piu+h)\ + \Sn.p{u+h)- Sn.p{u)\-\-\Sn,p{u)-S{u)\, the first and last terms on the right are, by (a), each < € / 3 for proper choice THEORY OF 8UMMABLE DIVERGENT SERIES. 27 of n, p, and the second term can be made < « / 3 by taking \ h\ < 5; hence \S{u+h) - S(u)\< e for I A| < 5, and our theorem follows. 00 31. Theorem 19. If the series zLciviu) is uniformly summable (III A) or (III B) with respect to u in (a, b) , with Sum S (u) , and if the terms a» (m) are integrable, then the series obtained by integrating the given series term by term toith respect to u over a range (cy, Ca ) included in ( o , b) is summable (III A) or (III B) respectively, with Sum I S (u) du. The proof of the part referring to (III A) is the same as that of Theorem 13. To prove the other part, let us write S {u) = Sn, p (u) + 5n. p (m), where 8n, p (u) approaches its limit lim 8n. p (u) = «,p->ao uniformly with respect to w in ( a , 6 ) . S{u)du= I Sn,p{u)du-\- I dn, p{u) du, and r[n] ^ Sn, p (w) du = 2/r (n, p) I a» (w) du, •=0 Jei By properly choosing n, p, we can make I 5n, p (W) I < € for every w in ( a , 6 ) ; K, p{u) du\<^ \ \hn, p{u)\\du\< e{c2 — Cx) for n, p properly chosen. .'. lim I 8n, p (u)du = 0, n,p— >oo t/cj F and [n] /»c, pct lim Ylfv(.n,p) I av (u) du = I S (u) du. n,p--^« v=0 Jci Jci 00 32. Theorem 20. IfY^a^iu) is summable (III A) or (III B) with Sum 00 S (u) , and if the series z2 a^ ( w ) obtained by term by term differentiation of the 28 LLOYD L. smail: some generalizations in the given series is uniformly summable (III A) or (III B) respectively, with Sum for every v, x; 2° lim/„(a:) = for t> fixed; OS 3** ^fv (x) is convergent for every x, and 00 \\m J2 fv (x) = 1 . » »->oe t»=0 The limit (12) cannot be used for the summation of non-convergent series (nor even for convergent series), for by 2°, n lim Ylsvfv (x) = 0, z— >ao t»=0 and therefore (120 lim lim 12sJ,{x) = 0. When the limit (13) exists and is equal to S , (130 lim lim J2s,fAx) = S, ae— >ao n—^oo v=0 we shall say that the series S a« is summable (IV) with Sum S . 00 34. Theorem 21. If the series ^a„ is convergent, vrith sum s, then n lim lim 2«t»/t» (x) = *; that is, every convergent series is summable (IV) with Sum equal to sum, so thta part (1) of the conditions of consistency is satisfied. Put Sn = s •{• 8n, where lim 5» = . THEORY OF 8UMMABLE DIVERGENT SERIES. 29 Then By 3°, so that Since llsvfv (x) = s • J^ fv (x) 4- Xl5„/„ (x). lim lim ^ fv{x) = 1 , x—^ao n— >« v=0 n n lim lim ^Svfv{x) = s -\- lim lim ^8vfv (x) . «— >w n— >oo »=0 z— >« n— ^00 v=0 lim 6„ = , we can find m such that | 5» | < e for c > m. Now n n H^vfvix) ^ Si 5j •/„ (a;), and •=0 r=)»»+l Now since for V finite, Z \8,\-fAx)00 lim Z/t. (a;) = 1, X— >oo and since « can be taken as small as we please, we have We have then lim lim Z 5» /» (x) = . w lim lim Z *» /» ( a; ) = * . 35. Theorem 22. If 22 civ is properly divergent, so that lim 5n = + 00 , lim lim Z^c/t. (a:) = + « ; /Acn part (2) o/ <^ conditions of consistency is satisfied. 30 LLOYD L. SMAIL: SOME GENERALIZATIONS IN THE As before, we can put Sv = K -\- r„ for v > m, Tv > 0, where K is an ar- bitrarily large number. Then Since «=m+l \imfr,(x) = 0, Hm lim Z/»(x) = 1, r„ > 0, fv{x)>0, z— ^00 s->ao n— >co we have n n lim lim ^s^fvix) = X + lim lim ^ r^fv {x) > K. X— >x n— >0B 11=0 i-^QO n— >ao ii=»»+l 00 36. Theorem 23. // the series ^av{u) is uniformly convergent in a closed n interval (a, b) , with sum s (u) , then 2J*» (u) fv (x) approaches its repeated double limit n lim lim 23 *« (w)/» (x) = s (u) X— >oo n— >co ti=0 uniformly with respect to u in {a, b); that is, a uniformly convergent series is aJ-so uniformly summable (IV), and part (3) of the conditions of consistency is satisfied. Write Sn {U) = S (U) -\- dn (U) , then we can find m such that for n > m , (a) 8»(w)| < € for every w in ( a , 6 ) . (6) lis, {u)f, {x) = s iu) ilfv (x) + 1:5, {u)fr, (x). 00 Now by 3°, zlfv (x) is convergent for all values of a; , so that if we denote its sum by F ( a: ) , we can find N such that ior n> N and for x fixed, Fix)-Zfv{x) < €. Then for any fixed w in ( a , 6 ) , we have F(x)'siu)-s{u)j:fvix) < €|*(w)| (n> N) THEORY OF 8UMMABLE DIVERGENT SERIES. 31 Now since s (u) is the sum of a series uniformly convergent in the closed interval (a, b) , \s {u)\ must have a finite upper bound, say K, in (a, 6); hence F(x) ' s(u)-s(u)T>fAx) N and for every w in ( a , 6 ) . Therefore s (u) ^ f^ (x) approaches its limit s (u) • F (x) uniformly with respect to w in ( a , 6 ) as n — > oo . Again, since by 3° , lim F (x) = 1 , we can find X such that for a; > X , \F{x) - 1| < €. For any fixed w in ( a , 6 ) , we then have \F (x) ' s(u) - s(u)\< e\s{u)\ for a; > X; therefore \F{x)siu)-s{u)\ X and for every u in (a, b) . Hence F (x) s (u) approaches its limit s (u) uniformly with respect to w in {a, b) as x-^ oo . n - Referring to our definition (§ 20), we see that s (u) ^fv (x) approaches its repeated double limit n lim lim s (u)^ f^ (x) = s (u) «— >ao n— >so uniformly with respect to w in {a, b) . n We have next to show that 21 5, ( m ) /„ (x) approaches its repeated double limit n lim lim ^dviu)f„ix) = uniformly with respect to w in ( a , 6 ) . We have T,8Au)fAx)-T.8Au)fvM By (a). ^E|5.(w)|/.(x) Sl+l E I 5. (m) I/, ix) < €Z/» (xX € Zf. (x) < e Z/. (x) ■»+l w+l for every urn {a,b) . But X/t> (ic) is convergent for every x, so that Fix)^tfvix) 32 LLOYD L. smail: some generalizations in the is bounded for all values of x, and if G be the upper bound of F (z) , we have i:\8Au)\fv{x) « , uniformly with respect to w in ( a , 6 ) . m is now fixed. Since lim/„ {x) = 0, for every v, we can find Xv such that for a; > X„, we have/r {x) < e; let X be the greatest of the finite set Zo , Xi , • • • , Xm , then we have each/^ ( a: ) < € fora;> X {v = 0,1, •••,m). Z8.(w)/.(ar) ^ll\K{u)\fAx)<€ll\K{u)\ {otx>X. m Now K{u) is bounded in (a, 6), so that Z^| 5» (w) | has a finite upper bound, call it H . Then for a; > X and for every w in ( a , 6 ) . Hence X)5„ (w)/t, (a;) approaches its limit , as ar — > 00 , uniformly with respect to w in ( a , 6 ) . n We have now shown that ^5» {u)fv (x) approaches its repeated double limit uniformly; and our theorem is now proved. 00 37. Theorem 24. If ^Ov {u) is uniformly summable (IV) loith respect e to u in an interval (a, 6) , with Sum S (u) , and if its terms a, (u) are con- tinuous in ( a , b) then S (u) is continuous in ( a , b) . The proof of this theorem can be carried through in the same way as that of § 23, the only difference being that here « we have n 00 Sx (u) = lim ^s„ {u)f„{x) = Y,Sv {u)fv (x) , n— >ao X SO that the series ^Sv {u)fv (x) is uniformly convergent, and we can in- tegrate term by term, and get ^fv{x) I Sr,{u)du= I Sx{u)du, the series on the left being convergent. Hence lim lim Z^fv {x) \ Sv{u)du = lim I Sx{u)du. Since limiS^(w) = S{u) and this approach is uniform with respect to w , by the method used in § 24 we can prove that Sx {u) du = \ S {u) du . -A •'Cl (a) .*. lim lim 2Z/t> (a?) I 5„(w)(iw= I S{u)du. «— >ao n— >oo t)=0 »/ci t/cj To prove that the series 2^ I a„ ( w ) c?m is summable (IV) with Sum I iS(w) dw, we must show that lim lim SiSI Ov (u) du\ f^ (x) = j S (u) du. a:— >oo n-^aa v=Q ^ p=0 tJci ' *Jc\ But zl \ av {u) du = \ •{^a„(w)f n, let /„ (n, k) = 0. We shall call this the Cesaro function, and denote it hy Cfv {n, k) . We must first show that Cfv{n,k) satisfies the conditions of § 6. It is easily seen that 3° and 4" are true for every k; also that 1° is true if k> 0, but if A; < and n> \k\, then CfAn,k)Sl. Since Cfv+i {n, k) n — V Cfv {n,k) k-\- n — v' this ratio is < 1 if ^* > , and is > 1 if A; < and n is sufficiently large, so that the sequence {Cfv) is decreasing for A: > and increasing for A; < and n large enough; 2° is then true for A; > . We shall not consider values of A: < . It remains only to show that 5° is satisfied. For Cesaro's method it is sufficient to give n only positive integral values. We need then only prove that lim Cfn(n,k) = 0. n—^aa We have* n ! n*' /I I (Ar+n)(ifc+n- 1) ••• (A;+ 1) ~ ^" ^^"^ ^ ^ ' * See Nielsen, Lehrbuch der unendlichen Reihen, p. 248. 35 36 LLOYD L. smail: some generalizations in the where limrn(A;+l) = r(A;+l). .'.\{mCfn(n,k) = 0. When a series S c^, is summable (I) with the Cesaro function Cfv (n, k) , it is usual to say that the series is summable (C , k) . The corresponding summation formula is rifi^ o _ ,. Y> n {n - 1) - • ■ {n - v-\- 1) ^ 42. The Theorems 1-3 give us the following results: Theorem 27. Every convergent series is summable {C , k) for every k > 0, ivith a Sum equal to its sum. This theorem has been proved by CHAPMAN.f Theorem 28. A properly divergent series is not summable ( C , k) with finite Sum for any value of k > . This theorem is new; it has been proved when A: is a positive integer, however, by Nielsen, t Theorem 29. A uniformly convergent series is uniformly summable {C , k) for k> 0. This has been proved by Chapman. § Riesz's Method. 43. Let the function /„ (n, k) be defined for, every positive value of n, for every real k , and for ij = 1,2, • • • , [ n ] , by the equation (17) y.(„,i)={i_Mil}'. where X (n) is a positive monotonic function of n, increasing to « with n; f or tj = , let fo{n,k) = 1, for this we must assume that X (0) = 0; and for r > n, let fv{n,k) = 0. We shall call this the Riesz function, and denote it by Rfv{n, k) . * When A; is a positive integer, this is the definition given by CesXro (BvUelin des sciences math., ser. 2, Vol. 14 (1890), pp. 114-120). When A; is any real number > — 1 , the method has been discussed by Chapman {Proc. London Math. Soc., ser. 2, Vol. 9 (1911), pp. 369-409), and by Knopp {Sitzungsberichte der Berliner Math. GeseU., Vol. 7 (1907), pp. 1-12). t Proc. Lond. Math. Soc., ser. 2, Vol. 9 (1911), p. 377. t Nielsen, Elemente der Funktionentheorie (1911), pp. 194-5. § Quarterly Joum. of Math., Vol. 43, pp. 24-25. THEORY OF SUMMABLE DIVERGENT SERIES. 37 It follows at once from the definition that conditions l*'-4'' of § 6 are satis- fied when ^* > . We shall suppose further that X ( n ) satisfies the condition : (18) lim — r-7 — r— = 1 , n— >« X(n) then it follows that and y X([n]) ^ hm > , . = 1 i^« X (n) n_>« L X (n) J so that condition 5° of § 6 is satisfied for A; > . When a series is summable (I) with the Riesz function Rfv (n, k) , Hardy and Chapman call it summable {R, X, k).* The corresponding summation formula is (19) S=limZa. l-fhri • n-»oor=0 «■ A(n)J 44. The Theorems 1-3 applied to this method give: Theorem 30. A convergent series is summable {R, X , k) for every k> 0, tcith Sum equal to sum, if\ satisfies the condition (18). This is proved by Chapman and Riesz. Theorem 31. A properly divergent series cannot be summable (R, X, k) , for any k> 0, with finite Sum, if\ satisfies (18). This result is new. Theorem 32. A uniformly convergent series is uniformly summable (R,\, k) for every k> 0,if\ satisfies (18). This theorem is also new. Case II. LeRoy's Method. 45. Let the function fv (t) be defined for all positive integral values of v , including , and for all values of t such that < t < 1 , by the equation T(vt+1) (20) /,(0 = r(^+i)- We shall consider it for values of t near 1. We call it the LeRoy function and denote it by L/„ (<) • This function is positive for all values oi v,t under consideration, and since ♦ See Chapman, Proc. Lond. Math. Soc., ser. 2, Vol. 9 (1911), p. 373. 38 LLOYD L. smail: some generalizations in the T (x) increases as x increases, for x > 1.462 • • • *, we have Also liinZ/„(0 = 1, and Lfo{i) = l. That the sequence Lfo, Lfi, • • • , Lfp, • • • is decreasing may be shown as follows: r(vi+t+l) T(vt+1) (a) LUUt)-LfAt) = r(2) + 2) r(2)+i) T{rt+t-\- 1) _ T{vt-\- 1) Now Tivt'\-l + t)= f e-'x'^+'dx, T{vt+l)= Ce-'x^'dx. By integration by parts, we get or I C-' «•*+' (ia; = <(» + 1 ) I e-' a:'"+*-i dx, Jo Jo .'. T (vt + t+ I) = t {v+ l)T {vt + t)< {v-\- 1)T {vt + t) , since <<1. But r (r/ + < r (r< + 1 ) , hence ib) T{vt+t-}-l)- {v+l)T{vt+l)<0. From (a) and (6), we see that Lfv decreases as v increases. We shall next show that hm ; 7; ; = o. By Stirling's formula,! we have r(n+lj ,^„e-''7i''+*v/2,r lim .v:;r;-^ =i, ... r(n<+ 1) e-*n'»+*/27r , . . lim ^^= • = 1 , »-»ooe-'»'(nO"'^l/27r r(n+l) * See the graph of r ( z ) in Klein, Hypergeometrische Funktion, p. 122, or Godefeot, Th6orie 616mentaire des series, p. 250. t See Bromwich, Theory of Infinite Series, p. 462. THEORT OF SUMMABLE DIVERGENT SERIES. 39 or T {nt+ 1) e-" n" • n* _ ^"^ iji r (n + 1 ) ' e-«' n"' n» r' v'< ~ ' Now * '^ ** _ g-n(l-0 yjn(l-Of-«« = g-n(l-0+n(l-0 log n-nt tog t = g(l-t)n log n-n [(l-0+< k« t] But since i < 1, (5) lim g(l-On log n-nll-t+t log <] = _|. qq ^ n— 3>as From (a) and (6), we see that we must have hm ^pTT TTV = . «_>« r (w+1) Then lim limi/rt (0 = 0. If we now make the change of variable t = e~^'' , we get a function of x: LfAx) = T(v-\-l) ' which satisfies all the conditions of § 13. When a series is summable (II) with the LeRoy function Lfv (x) ,we shall say that the series is summable (L) . The corresponding summation formula is (21) g = limlim± %7;'+^> . This definition is one used by LeRoy.* 46. When we apply the general Theorems 7, 8 to this method, we have the theorems: Theorem 33. A convergent series is summable (L) mth Sum equal to sum. This was proved by Hardy, t Theorem 34. A properly divergent series is not summable (L) with finite Sum. This result is new. BoreVs Method. 47. Let the function fv (x) be defined for all positive values of x , and for every positive integral value of v , including , by the equation (22) BfAx) = f'e-f^da, ♦ Annalea de Toulouse, ser. 2, Vol. 2 (1900), pp. 323-327. t Quarterly Joum. of Math., Vol. 35 (1903), pp. 36-37. 40 LLOYD L. smail: some generalizations in the This function is always positive; and since the integrand is positive, we have rg~" a" da < I e~* a" da = D ! 80 that Bfv (x) < 1 for every v, x. Thus condition 1° of § 13 is satisfied. By integration by parts, we find /c— a*+^ da= - e-^ a"^^ + {v -{- l)f e'' a" da, •*• / ■ IN , e— a*^^ rfa = - , , ,,, + -i e-^a^'da, (»+ 1) !Jo {v+l)l vlJo or 5/h.i (a:) - 5/. (x) =- e--— ^ (1^+ 1)1' The right hand member of this equation is always negative, so that as v increases, Bfv (x) decreases, and 2** is satisfied. We have at once lim5/„(a;) = 1, and 3° is satisfied. That 4° is satisfied, may be shown as follows: We first seek the limit Stirling's formula gives us Now a" e-" n"-^ 1^2^ a* lim — , . hm ^. . — = 1 . g-n „«+j y/2ir n 1 n 1 e'" n""'"* /2ir * so that a" . a" (a) lim — , = lim ^. , — . We have a"" g—n ff^n+i —. gn ^n ij— »»— 4 =: gw+w •<>« «— (n+1) log « ^ g"'^"'" •** •!— (»»+4) tof « But lJ|jl'gn{l+IOga)-(n+J)IOgn _ Q (6) .Mim^=0. THEORY OF 8UMMABLE DIVERGENT SERIES. 41 We can then find N such that a" / n I < e for n > N . .•.jrc-^da< €j e-^da= €{l - e"') f orn > N. .'. lim I e~^—.da = 0, m— >ao »/0 W I and lim limfi/, (x) = 0, SO that 4° is satisfied. All the conditions of § 13 are therefore satisfied by Borel's function. The summation formula for this method is S = lim lim Yl(h • \ e~^—:da = lim lim 1 e~^i^av—.)da. If the series 22(h> oc^ I vl defines an integral function, or if it defines an analytic function which is susceptible of analytic continuation along the real axis, and therefore has no singular points on the real axis, we get S = lim I e~'^[^av—.)da. z-^co Jo \ '^ • / (23) .,s = J".-.(|:a.f,)ao X(n) It is easily seen that conditions 1°, 2°, and the first part of 3° of § 26 are satisfied. Hardy and Chapman t have shown that the second part of 3° is satisfied. Since we have '^ .(.) so that 4" is satisfied. Since hm 1 1 — , . . f = r^^ L X (n) J lim lim /t, ( n , p ) = 1 , p->oo n—^co X(n-l) hm ^ . . — = 1 , n->oo X(n) and X(n)— > « asn-» oo,we have X(") so that 6° is true. «->« l X(n) J * Called thus by Hardy and Chapman, Quarterly Joum. of Maih., Vol. 42, p. 191. t Quarterly Joum. of Math., Vol. 42, p. 204. THEORT OF 8UMMABLE DIVERGENT SERIES. 43 Now suppose that the path F is defined by the relation = w ( n ) , where lim w( n ) is finite; then "•VI X(n)J „^« I X(n)J *'V I X(n) J «^, I X(n) J Thus 5° and 7° are also satisfied. When a series is summable (III A) with the Cesaro-Riesz function, we shall call it summable ( CR , X ) , and when it is summable (III B) and the path F is given by the function w ( n ) , we shall call it summable ( Cil , X , « ) . 51. From Theorems 15 and 17, we have: Theorem 37. A convergent series is summahle ( CR , X ) * toiih Sum equal to sum. This theorem is new. Theorem 38. A convergent series is summable {CR, X, co) for every path F {as described in § 50), mith Sum equal to sum. This was proved by Hardy and Chapman.! Theorem 39. A properly divergent series is not aumvable {CR, X ) nor {CR,\,o}). This result is new. Case IV. 62. Let us take first (27) fAx) = e-^ 'f-^. We have at once fv {x) > , lim /„ ( a: ) = , Zfv{x) = e-'f:^^=e-''e'=V, «' ' so that conditions V, 2°, 3°, of § 33 are all satisfied. The corresponding sum- mation formula is (28) S = lime-'E*.-;, which is Borel's exponential definition. J When a series is summable (IV) with Borel's function (27), we shall say that it is summable { Bi) . * Where X satiafiea the conditions of § 50. t Quarterly Joum. of Math., Vol. 42, p. 204. X See BoBEL, Legons sur lea series divergentes, p. 97. 44 LLOYD L. 8MAIL: SOME GENERALIZATIONS IN THE More generally, take (29) /.(a:) = 6-*.^*, where A: is a positive integer; this can be shown to satisfy the conditions of § 33 just as for (27). When a series is summable (IV) with Borel's function (29), we shall call it summable {Bz, k) . When k = 1 , we have summability {B\) . The summation formula is (30) S= lime-* £5,^. This is Borel's generalization of his exponential method.* If we take (31) /.(a:) = e-'.^, it is easily shown that this function satisfies the conditions of § 33. The resulting method is one studied by CosTABEL.f Still more generally, let {x) be an integral function 2Z CnX^ , where Cn > . Take (32) /«(^) = ^ •^'^^- This function satisfies the conditions of §33, and includes all the preceding functions of this §. 53. Applying Theorems 21, 22 to Borel's methods, we have: Theorem 40. A convergent series is summable {B\) with Sum equal to sum. This is proved in Vivanti-Gutzmer, Theorie der eindeutigen analytischen Funktionen, pp. 328-9. Theorem 41. A properly divergent series is not summable (Bi) toith finite Sum. This also is given in Vivanti-Gutzmer, p. 329. Theorem 42. A convergent series is summable {Bz, k) with Sum equal to sum for every k . This result is proved in Bromwich, Theory of Infinite Series, pp. 300-301. Theorem 43. A properly divergent series is not summable {Bz, k) toith finite Sum, for any k . This is new. 54. The general theorems of Chapter II on the uniform summability of uniformly convergent series, and the continuity, integration, and differentia- tion by uniformly summable series apply of course to the particular methods of this chapter, but as all the results so obtained are new, they have not been stated explicitly. * See BoREL, Legons sur les series divergentes, p. 129. t L'Enseignement mathimatique, Vol. 10 (1908), pp. 387-388. REFERENCES. BoREL, E. Le9ons sur les series divergentes. Paris, 1901. Bromwich, T. J. Ta., Introduction to the Theory of Infinite Series. London, 1908. (Chap. XL) Cesaro, E., " Sur la noiultiph cation des series," Bull, des sc. Math., ser. 2, Vol. 14, pp. 114-120. (1890.) Chapman, S., "On Non-Integral Orders of Summability of Series and In- tegrals," Proc. London Math. Soc, ser. 2, Vol. 9, pp. 369-409. (1911.) Chapman, "On the General Theory of Summability, with Applications to Fourier's and other Series," Quarterly Journ. of Math., Vol. 43, pp. 1-52. (1911.) CosTABEL, A., " Sur le prolongement analytique d'une fonction m^romorphe," VEnseignement Math., Vol. 10, pp. 377-390. (1908). Hardy, G. H., "On Differentiation and Integration of Divergent Series," Tram. Cambridge Philos. Soc, Vol. 19, pp. 297-321. (1903.) Hardy, " Researches in the Theory of Divergent Series and Divergent In- tegrals," Quarterly Journ. of Math., Vol. 35, pp. 22-66. (1903.) Hardy and Chapman, " A General View of the Theory of Summable Series," Quarterly Journ. of Math., Vol. 42, pp. 181-219. (1911.) Knopp, K., " Multiplication divergenter Reihen," Sitzungsherichie der Berliner Math. Ges., Vol. 7, pp. 1-12. (1907.) Le Roy, E., " Sur les series divergentes et les fonctions definies par un d^ veloppement de Taylor," Annales de Toulouse, ser. 2, Vol. 2, pp. 317-430. (1900.) Nielsen, N., Lehrbuch der unendlichen Reihen. Leipzig, 1908. Nielsen, Elemente der Funktionentheorie. Leipzig, 1911. RiESZ, M., " Sur les series de Dirichlet," Comytes Rendus, 148, pp. 1658- 1660. (1909.) RiESZ, " Sur la sommation des series de Dirichlet," Comptes Rendus, 149, pp. 18-21. (1909.) RiESZ, " Sur les series de Dirichlet et les series enti^res," Compies Rendus, 149, pp. 909-912. (1909.) Vivanti-Gutzmer, Theorie der eindeutigen analytischen Funktionen. Leip- zig, 1906. 45 VITA. Lloyd Leroy Smail. Born, Columbus, Kansas, September 23, 1888; entered University of Washington, 1907; Tutor in Mathematics, 1909-11; Undergraduate Assistant in Mathematics, 1910-11; A.B., 1911; Fellow in Mathematics, 1911-12; A.M., 1912; Fellow in Mathematics; Columbia University, 1912-13; member of Sigma Xi; member of the American Mathe- matical Society. The writer gratefully acknowledges his indebtedness to Professor W. B. 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