ON THE DEFINITION OF THE SUM OF A DIVERGENT SERIES 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 UNIVERSITY OF MISSOURI COLUMBIA, MISSOURI APRIL, 1913 PRESS OF THE NEW ERA PRINTING COMPANM LANCASTER. PA. CONTENTS Page ~ I. INTRODUCTION.................................. I ~ 2. HISTORICAL RESUME............................. 3 ~ 3. AVERAGEABLE SEQUENCES........................ 15 ~ 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 v ~ I. INTRODUCTION * The series u0 + ul + u2 + *** is defined to be convergent whenever L (uo + ul + * * + n u) 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) = oo; all series which neither converge to a finite limit nor diverge to infinity are then called oscillatory.t The present considerations are limited to series which are oscillatory. We shall follow, however, the terminology of most writerst 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 Un = 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 fundamental properties of convergent series. We shall, in fact, demand the following fundamental requirements of any generalized 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, I9Io, in partial fulfillment of the requirements for the degree of Doctor of Philosophy. t Bromwich: An Introduction to the Theory of 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 Uo + u1 + U2 + * * U1i + U2 + *. has a generalized sum, whenever the other has, and t = s - uo, if s and t are their respective sums. (iv) If each of the series I + U + U 2 + U + ** V0 + V1 + V2 + ' * has a generalized sum, A and B respectively, then the series (uo + Vo) + (ul + vl) + (u2 + v2) + * has a generalized sum which is A + B. (v) If the series uo + ul + u2 +... has s for its generalized sum, then kuo + ku1 + *. 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 indebtedness to him for many helpful and important suggestions. I am also indebted to Drs. W. A. Hurwitz and H. M. Sheffer for many suggestions and criticisms. ~ 2. HISTORICAL RESUMEF * 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 - s. We have then, - I = - or s = ~. 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 considerations. 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, 2. This same value was later attached to the series by Euler, 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 + 2 -3... will be I /(I + x), because the series arises from the expansion of the fraction, whatever number is put in place of x."~ In particular, = I - I + I -I + * The best historical sketches are to be found in Borel: Lemons sur les Series Divergentes: Introduction, and in an article by Pringsheim given immediately below. t See Pringsheim: Encyclopddie der Math. Wiss., I, I, p. I07, note. t Instit. Calc. Diff. (1755), Paris, II (p. 289). ~This 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 I -x2 I + X + I2 - X - I - X2 + X3 - X5 + X6 - X +... I + X + X2 I - X3 and setting x = I, 2 = I - I + I - I + *. To meet this difficulty, Lagranget observed that we should write =I + X - = + o ~ x - X2 + X3 + o. X4 - X5 +..-, I + X + X2 so that for x = I, we have =I+O —+I +o-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 accordingly 2. 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, which is a generalization of Abel's theorem: THEOREM A:~ If Sn = Uo0 + U 2 + 2 + + + n and I o + S + * ' + Sn+ * By Callet. See reference immediately below. t Rapport sur le Memoire de Callet, in: Memoires de la classes des Sciences mathematiques et physiques de l'Institut, 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 uxn = s. Z=l 0 Thus, in the case of the series I I + I -I -+ *, So + S1 + * * + Sn I nL - 2=' =and a y 2 and accordingly L (i - x + x2.) =; so that we may define the value of the series I-I + I..- to be L ( - x X=-1 + x2 - 3 +...), or what amounts to the same thing, Lso+ S + s ' * + S nX n whenever the limit exists. The first mathematician actually to carry through the definition was CesAro,* who approached the subject from another standpoint. Cauchy has defined as the producttof two series I Uo + U + Vo + vI + * * * the series uovo + (UoVI + UiVo) + (uov2 + UiVi + U2Vo) + *.; 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. Mertens has generalized this theorem by proving that the Cauchy product of an absolutely convergent series by a simply convergent series is convergent. The product of two simply convergent series may, however, be divergent. CesAro 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 mathematiques, t. XIV, I890. 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 Uo + U1 + U2+ * Vo + Vl + V2 + *.. converge to u and v respectively, and let Wn = (UoVn + UlVn-1 + * * + UnVO) Sn = Wo + Wl + * * + Wn then So +s1+ S + ** =+ Sn L = u v. n=ao n + I The two theorems which we have stated justify us in stating the following definition: Definition:* If s, = uo + Ul + u2 + * + Un, the series uo + ul +... + Un + * * is summable and has the value s whenever so + Si + * * * + Sn L- -"S- =s. on= 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 = uo + ul + *- * + un, and L Sn = s; we shall prove n=00 that so+sl + S1 + * * * + s L S. = n= o, n + I We have: 0+So i+. + + sn ~~n + i | (So-S)+(Sl-S) + * * * +(S-S)+(S+-S)+ * * * +(S.-S) _~ n + i _< -so-s i+Is,-sl+ +|lss_-s l + |S~-sl|+ * +|sn-sl n+I n + I * CesAro calls series of this type simply indeterminate. t By this theorem requirements (i) and (ii) are satisfied. DEFINITION OF SUM OF A DIVERGENT SERIES 7 Since L Sn = s, we can take q so great that I si- s < e/2, i n=c i ~ qg. Having chosen this g, let L be the largest of the numbers, Si -, i = O, I, 2, g - I. Then we obtain: So + S1 +. + Sn qL (n-g + I)e qL e n + I -n+ I 2(n+ I) n + I 2 We can now choose n so large, n > r, that qL e n+ I <2 and hence, So0+ + 1 + * * + Sn o- — n +- I s <e, n >r. ' ' =so+s + + s. n-e _ n + I THEOREM D: * Each of the series J uo + U + U2 + U * + Ui + U2 + * * 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, s1, * sn,... is summable and has s for its sum, then the sequence Si, s2, * * * S, * * * is also summable, its sum being likewise s. For, S1 S2+ S1i + 2 + + 0 l + + Sn+ L L= - + L + 1n2 n +I n=~n n + n= n + I So + S + '' *+ S- n+1l S o + + ' + Sn+lI n + 2 L L n.oo n + I n=o n+2 n+I So + sL + + +Sn = L = s. n=oo n + I * By this theorem requirement (iii) is satisfied. 8 UNIVERSITY OF MISSOURI STUDIES To return now to Theorem D; we wish to prove that if uo + ul + u2 + - * is summable to s, then ul + u2 + *- is summable to s - uo. The sequence corresponding to the series O + Ul + U2z + * is uo, uo + U1, *. By the lemma proved above, it follows that the sequence Uo+ui, uo+u1+u2, * *. or s1, 52, -. is summable to s. The sequence corresponding to Ul + U2 + * is u1, Ul + u2, *. which may be written Si - Uo, S2 -, *' '. Now THEOREM E:* If u + uI+ U -' v o + Vi + * * are summable to u and v respectively, then the series (uo + Vo) + (u1 + v1) +..* is summable to u + v. Writing s, = uo+U + Ul + + * + n, nt = vo + Vl +.. + vn, we have + t = (u + )(U ) + (... + ) + (u, + vn). We obtain: (So + to) + (s1 + tl) + * + (sn + tn) n=o n + I So + 1 " + Sn - to + tl+ + t ' + tn nL+ + L = n+v. n=c n + I n=o n + I Cesaro'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 definition 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 Uo + ul + * + u,n * is Un L 0. n=-o = Since the series is summable, Ls~+s~+'+s- L So+Si+Sn so + s + * + s~ So + sl + ' + s n nC=oo n ~n=o n + I so + S1 + * * * + Sn- 1 So + Sl + * * + Sn = JL s L n= co n n=co n Sn = - L = o.,=co n Hence: Un - - n- -- Sn Sn —1 L- = L = L -- = o. n=fo l n=co n n=ao /g n=lo T We are accordingly limited to series for which (I) L =o. nr=oo n But such a simple series as I-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 x = I in the expansion: (+ )2= I -2+ -X2., (I + X)2 we obtain = I-2+3-4 4 +. We are thus led to extend, with Cesaro, the above definition of summability of order I, to summability of order 2. We say that a series is summable of order 2, if L (n + I)So + nsl + * + 2Sn-l + Sn S 7n a (n + I)(n + 2) 2 IO UNIVERSITY OF MISSOURI STUDIES A necessary condition* for the existence of this limit is that Un L =o, n=o n so that we cannot evaluate the series, r(r + I) r(r + I)(r + 2) I-r+~ -- + *"' r > 2, 3! although we obtain by Euler's method, I r(r + I) (r + i)(r + 2) (= x - rx+ X2 - 3 x +... (I + X)r 22! 3! and accordingly I r(r + I) r(r + I)(r+ 2) -I-r+!- +.... 2't 2 2! 3! We are thus led to state the following more general definition: Definition:t The series u0 + U + u + u * * * is summable of order r, if r is the smallest integer for which there exists the limit: r(r+I).. (r+n- ) r(r+I)* (r+n-2) sI n! (n-I)! r(r+I) +Sn-2 - +Sn-lr+Sn (2) L () o (r+I)(r+2)... (r+n) 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. Let us now return to CesAro's first definition, and observe that we may generalize it in a more natural way. * Bromwich, loc. cit., p. 318. t CesAro, 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 t (i) _ so + + * * * + t n ~in + I tet smlt n~I I t<(ri) ^to(r) q_ t1(r) 4-... _ t() tn r+l) ' r = I, 2, then the smallest integer r for which L t~(r) exists, shall make the series summable of order r. To distinguish this definition from that on page Io, we shall call the definitions CesAro-summability of order r and Holdersummability of order r, denoting them briefly by (Cr) and (Hr) respectively. It is knownt that these two definitions are equivalent 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), thent Un L = o. n=co n Accordingly the series I- t + t2 - t3 + * (t > I) does not have a sum (C,) for any value of r; since L to, t I. n=O nr We are thus led to generalize still further the definition for the sum of a series. From the definition given on page Io, it is clear that we may write Cesaro's forms as follows: L aOSo + aiS1 +.. + anSnl s=cn L 0ao+asl+ +an J' * Holder: Mathematische Annalen, Bd. 20, p. 535. t Schnee: Math. Annalen, Vol. LXVII (I909), p. IIo. Ford: Am. Journal of Math., Vol. XXXII (1909), p. 315. t Borel, Series divergentes, p. 92. 12 UNIVERSITY OF MISSOURI STUDIES where the ai are functions of both n and r, r being fixed.* Let us choose as our definitiont L L [ ao(r)so + a(r)s + ' "+ a(r)s,,] r=o no ao(r) + al(r)+ *..+ a,(r) In particular we shall take a,(r) = rP/p!, and obtain r r2 rn so + si - + s2 ~ + + Sn s= LL - 2 --- r=co n=oo r r2 (4) [ r+~ +i +. (4) I +2!+** n! = L L e-r 0 + 1 - + * * * + Sn \ r=oo n==ao I 1 It can be proved readily: that this limit exists, whenever the series converges. We shall now transform~ this limit. Let 11 r f2 rn s(r) = So + SI + S2 + * * + Sn + * *, r r2 rn s'(r) = S1 + S2 + S- + *** + Sn+l + **, then r r2 yn uI(r) = s'(r) - s(r) = U1 + U2 + U2 - + *. + Un n + * * *. But -[e-s(r)] = e-r[s'(r) - s(r)], dr so that e-rs(r) = e-r[ s(r)- s(r)]dr + to and co s - Uo = e-ru(r)dr. * Borel, Series divergentes, p. 94. t r is now a positive real number. t Bromwich, loc. cit., p. 298. This is a special case of Th. 12, p. 52 (infra). ~ Borel, loc. cit., p. 97. I It is assumed that s(r) is convergent for all values of r; otherwise the limit (4) would have no meaning. DEFINITION OF SUM OF A DIVERGENT SERIES 13 If now we integrate by parts we obtain: s - Uo = [e-r f u(r)dr] + f e[r f u(r)dr]dr, or, if we let: r2 rn r u(r) = uo + ulr + u 2. + * u + * - = Uo + Uu(r)dr, s -u = [e-r{u(r) - u0}]o + e-r[u(r) - uodr' =[e-ru(r)]o - u [e-r]o + e-ru(r)dr- io e-rdr = [e-u(r)] + e-u(r)dr,. e., - uo = L [e-u(r)] - U + e-ru(r)dr, or s = L [e-r(r)] + J e-u(r)dr. If now we assume* that e-ru(r)dr is convergent, then it follows from the last equation that L [e-ru(r)] must exist. But this r=co limit must necessarily be zero, for otherwise, the integral would not converge. Hence we obtain r r2 rn where u(r) = U + u + u + u2 *+. +un- + **, whenever the integral converges. It can be provedt here, too, * We have gone into greater detail here than does Borel, loc. cit., p. 98. But this is essentially his argument. t Bromwich, loc. cit., p. 269. 14 UNIVERSITY OF MISSOURI STUDIES that when the series uo + ul + *- + u + + - converges, so does the above integral, and their values are the same. Furthermore Borel proves the following theorem: THEOREM G:* If the Borel-integral definition t applies to the series: U1 + U2 + *' + Un + *' = S, then it also applies to the series Uo + u1 + u2 + * *, giving for its sum s + Uo. The converse, however, is not necessarily true. Thus if the series uo + ul + u2 + - * is summable by (5), it does not followl that the series u1 + u2 + * - 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 Uo + ul + u2 + *... shall be called absolutely summable, whenever the integrals e-r u(r) dr, ao e-r I u()(r) dr converge, where X denotes the order of any derivative. That this definition satisfies requirement (iii) is proved by the following theorem: ~ THEOREM H: If either of the series Uf o + + U2 + * lut U1 + U2 + '* is absolutely summable, so is the other; and if s, t be their respective 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.[I * Borel, loc. cit., p. IOI. 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: Annales de la Faculte de Sciences de Toulouse (2~ series), t. 2 (I902), p. 317. See p. 60, footnote. ~ 3. AVERAGEABLE SEQUENCES On page 4 we have considered the series I- I + I - I + I + 0 - I + I + 0 - I +..-, and, replacing them by their respective sequences, we obtained = I, o, I, o, *. -1 = I I0, O I, I, 0, * 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 Frobeniust and Cesaro,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, of ** *, it is not easy or even possible to state the multiplicity of the limit-values in general, as is evident from the following example: sieo, i t+n2 } Si = o, i = n2I t= n, So, S1, S2,... Sn,.... 2 fl 0, I,2, To meet this difficulty, we shall proceed as follows. Let us assume, to be concrete,~ that the sequence So, S1, S2, * ' Sn, * * See page 3. t See page 4. t See page 5. ~ We shall go into every detail in only this simple case; the later generalizations we shall outline only briefly. I5 I6 UNIVERSITY OF MISSOURI STUDIES has two limit-values 1l and 12. Then we have Ism - lil < e, ISn - 121< 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 si which fall in the interval about 11, and n which fall in the interval about 12. Since m and n are functions of i, we may write m = fl(i), n = f2(i). If we choose e sufficiently small, and i > N, we shall have fl(i) +f2(i) + k =i, where k is a constant independent of i. Definition: The sequence se, S1, s2,.* s,, * *, having 11 and 12 as limit-values, shall be called averageable and have s for its sum provided L f(i)l l+ f2(i)12 iLm. fl (i) +f2(i) J s That this limit, when it exists, does not depend upon the particular e we have chosen follows at once. For if we take e < e, calling the corresponding functions fi(i) and f2(i), it is clear that fl(i) = fl(i) + k f2(i) = f2(i) + k2 where k1, k2 are independent of i. We accordingly have: [fl(i)ll + f2(i)12 1 [fi(i) - k]l + [f2(i) - k2]12 t - L fl(i) +f]2(i) J = l [fi(i) - k,] + [f2(i) - k2 fi(i) - ki, f2(i)-k2 fl(i) - k1 f2(i) - f2i)12 i==0 fi~) - i fiW + 2(i)- k2f2(i) i fiL(i) f2(i) L [fl(i)ll + f2(i)121,=L fl(i) + f2(i) ]' DEFINITION OF SUM OF A DIVERGENT SERIES 17 since kl k2 LJ L e - o. i=-fl(iW i=f2(i) Let us now find the sum of the sequence suggested on page 15, I; O, o; I, O, O, O, O; I, O, O, O,,,; ', i. e., Si =, i =n2 1 =o, i n2J Let us choose i = m, and let n2 be the largest square integer less than or equal to m. Then we have: s L n. I + (m - n) o L n M=ao m m=x m since n2 < m. Let us now see whether this definition satisfies the requirements of page 2. The first two requirements are obviously satisfied. As to the third, we observe that corresponding to the series Uo + Ul + U2 + * * * + un + * *; u1 + U2 + * * + Un + - *, we have the sequences s0, sl, S2,.. Sn, * '; S1-Uo, S2 - Uo, *~ s, - uo, *.-; and if the limit-values of the first sequence, which will be assumed to be averageable to s, be 11 and 12, then those of the second sequence are 11 - uo, 12 - io. We accordingly have: [fl(i)[lli- Uo]+f2(i)[12-uo] [fi()l+f(i)12] We shall now show that the fourth requirement is satisfied. THEOREM I: The sum of two averageable sequences is itself averageable, and has for its value the sum of their respective values. Let the two sequences SO, 51, S2, ''' Sn, ' to, tl, t2*,. tn,. I8 UNIVERSITY OF MISSOURI STUDIES have 11, 12 and ml, m2 as their respective limit-values, and s and t as their respective sums. Then we have: =L [fl(i)li +f2(i)121 = L L fl(i) + f2(i) J [gi(i)mi + g92(i)m]2 i= L [gl1(i) + g2(2) ] We wish to show that the sequence So + to, S1 + tl, * * Sn + tn, * * is averageable, and has for its value s + t. We observe that the only limit-values for the sum-sequence are l1 + ml, l1 + m2, 12 + mi, 12 + m2. Let us call Fij(n) the number of the (si + tn) which are near the limit-value li + mj. Then we have to consider:* [ Fll(n)(1l+ml)+ Fl.(n)(11+m + F21(n) (12+mO) [FL W~h~^+F~n )+F q+ +F22(n)(12-+m2) n-; Lo Fil(n) + F12(n) + F2l(n) + F22(n) It is clear, however, that F1l(i) + F12(i) = fl(i) + cl Fll(i) + F21(i)= gl(i) + dl F21(i) + F22(i) = f2(i) + C2 IJ F12(i) + F22(i) = g2(i) + d2 where c1, c2, d1, d2, are constants independent of i. We accordingly obtain: Fll(n)(11 + mi) + F12(n) (11 + m2) + F2l(n)(12 + mi) LIY~~ +F2 2 ________q+ F22(n)(12 + m2) n-L0 Fll(n) + F12(n) + F21(n) + F22(n) [Fl1(n) +Fl2(n)(]l+[F2l(n) +F22(n)]l2 ] F l() +[Fl(n)lln) +F21(] + [F2+n) 2n)+F22(n)]m n= l Fl (n) + F12(n) + F21(n) + F22(n) * We have defined averageability for sequences with only two limit values. The extension to sequences with any finite number of limit-values is obvious (see page I9). DEFINITION OF SUM OF A DIVERGENT SERIES 19 - [fl(n)+Cl]ll+ [f2(n)+C2]12 fl(n)+cl+f2(n)+c2 [gi(n) +dllml+ [g2(n) +d2lm2 n=o gi(n) +dl+g2(n) +d2 -L [fl((n)l + f2(n)l12 r gl((n)ml + g2(n)m2] -n= L f(n) + f2(n) nL gL(n) + g2(n) s + 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 limit values is obvious. Definition: A sequence having k limit values, 11, 1?,.* * k, shall be called averageable, and have s for its value, if n=k Efn(i)ln n=l 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: (E) (E(~)) =so S, S,2, * Sn, (E(1)) - lo(') 11(1) 2(1)I... n(l)... (E(2)) 1 —o(2), 11(2), 12(2),... n (2),... (E(k)) _ 10(k), 1 (k), 2 (k),... (k)),..., where the sequence (E(i)) consists of the limit values of the sequence (E(i-1)). Since the sequence is assumed to be reducible, there exists a k such that (E(k+l)) = o. Then (E) is reducible of order k, and (E(k)) 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 (E(k)) has for its elements /(k), Il(k), * * * I(k). If now we choose e sufficiently small, all but a finite number of the li(k-1) will fall in the intervals li(k-) - Ip(k) I < e, p = o, I, 2, *. P. Suppose that the finite number of 1i(k-l) which do not fall in any of these intervals is pi, and call them, m(k-l), m2(-l), *. mp1(k-l). We can choose el < e, so small that only a finite number, P2, of the li(k-2) do not fall in any of the intervals above, or in the intervals li(k-2) - mp(k-l) < el, p = I, 2,. *. Pi. Call this finite set of limit points mi(k-2), * m,(k-2). We can repeat this process until we reach the sequence (E), which will have only a finite number of elements outside of all the intervals considered. Definition: A reducible sequence shall be called averageable, with s for its sum, provided* j=k ipk-j+l E E f' j(n, e)mi() ~s = =~L L jl=k= $ L F(e) = fi(j)(n, e) e-o j=l i=1 exists. In this general definition it is convenient to distinguish between different kinds of limit points. Let us suppose that fi(n, e) corresponds to the limit point mi, and let us assume that the following limit fi(n, e) ai = -L jk i=Pk-j+l En=o Z E fi(n, e) j=l i=1 exists for every i. We shall call mi a weak or a strong limit point according as ai is or is not equal to zero. We may then state the following proposition: THEOREM 2: A reducible averageable sequence with a finite number of strong limit points is averageable independent of e. * We have put mi(k) = i(k) for the sake of uniformity. DEFINITION OF SUM OF A DIVERGENT SERIES 21 For simplicity consider the case where the reducibility is of order 2. The strong limit points are then either of the first or of the second order. There is only a finite number of strong limit points of order 2, and a finite number of strong limit points of order I. Let m be the total number of strong limit points. Since for the remaining limit points ai = o, we have n= fi(n, e) If we now choose e' < e, the values of the coefficients of the strong limit points are unaffected. Hence F(e') =F(e), and our theorem is proved. THEOREM 3: A reducible averageable sequence with a finite number of strong limit points is Cesaro-summable of order I; and the two values obtained are equal. We lay off ei intervals about the limit points of order k-i-I, (i = I, 2, *. k) as on page 20, and we thus have for n > N, if e is the largest of the ei, i l- Si' I <e, i =, 2,...fl(n, e) where s.(i) are those si 12 - si" I < e, i = I, 2, f2(n, e) which fall in the e-interval about ji. I I-si(P) < e, i = I, 2, fp(n, e) We have accordingly: I (fill +f212 +.* + fplp) - [(Si' + * * + Sf) + * * + ((S1() + * * + Sfr())] < (fl +f2 + fp)e. Since (S/' + S2 + ) + + (S(P) + * * + Sfp(P)) = Sm+1 + Sm+2 + * ' + Sm+, where q =fl +f2+ +'" +fp, and m is sufficiently large, we have: 22 UNIVERSITY OF MISSOURI STUDIES fill + f212 + ' * + fplp Sm+l + Sm+2 + * + Sm+q < I g g q q Hence L [fll + f22 + * + fplp] qL [Sm + Sm+l+.. + Sm n-=-, ~l + ~2 * -- + fpqL 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 (C1). 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 Si, Si+k, Si+.2k, * converges, then the sequence sl, s2,... is summable (C1). Let us take as an example the sequence Si = i log ( I + ), i odd =, i even to which it is not easy to apply the formula L SI + S2~ + sn n= — n We see, however, that the two sequences SI, S3,.* * * S2, S4, J converge; hence the given sequence is summable (C1). ~ 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 s and t are the respective sums of two convergent sequences, SO S, S 2, '' * Sn, * * tO, t,, t2, * *.* tn, ***, then it follows from a fundamental theorem of limits that L snt, = st. n=00 We are accordingly ledt to propose the following Definition: The natural-product of two sequences, So, S1, s2, ' Sn,. '; to, tl, ' * * tn * *', is the sequence: soto, sltl, *.. Sntn, * t 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 theorems for the Cauchy-product, it will be seen at once that the naturalproduct 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 naturalproduct can play no part. For consider the simple example, * See page 5. t Baire: Cours D'analyse, t. I. t Theorem B, page 6. 23 24 UNIVERSITY OF MISSOURI STUDIES S = I,, I, O, *t = I, I, 0, *. W = I, 0, I, 0,, 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 = I, 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 So, S1, S2, ' * Sn, ''* tO th, t 2,. *. tn, *.* and let us form the array: soto, Sots, Sotb, ' SOtn, Slto, sltl1, slt2, * ' - Sitn, ' S2t, S2tl, S2t2, * ' S2tn, * * l *. * * *. * * *. SntO, Snt, Snt2, ''' Sntn,.............................................................. Definition: The sequence formed by following the successive lines which form squares with the boundaries of the array, i. e., Soto; Sot1, s1t1, sto; SOt2, st2, S2t2, S2tl, S2; S *, 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 S SO, S, * * * Sn, t = to, t1,.* t, *; we wish to prove that the sequence soto; sot1, s1t1, silo; sot 2, s 2t2, S2tl, S2t0; *' is averageable, and that its value is st. We shall assume* that the sequence (s) has the two limit-values 11, 12, and that the sequence (t) has the two limit-values mi, m2. The only limitvalues of the product sequence are then: 11ml, 1lm2, 12ml and l2m2. We are given fl(n)ll - f2(n) 12 s= - fl(n) +f2(n) ' t = L [gi(n)1I + g2(n)2 1 n=L gl(n) + g g2(n) and we wish to consider: rL Fll(n)lml + F12(n)l1m2 + F21(n)12ml + F22(n)12M2 n -l(n) + Fl1(n) + F21(n) + F2F(n) +F n) where Fij(n) is the number of elements of the product sequence near limj. If we pick n elements from the product sequence, we observe: Fll(n) = fl(n)gl(n) + kll F21(n) = f2(n)gl(n) + k2 Fl2(n) = fl(n)g2(n) + k12 F22 (n) = f2(n)g2(n) + k22, where kij are constants independent of n. We have, accordingly, L [ Fll(n)llml + F12(n)llm2 + F2l(n)12ml + F22(n)lm2] nLF= Fll(n) + F12(n) + F21(n) + F22(n) [fi(n)gi(n) + klllllm + [fl(n)g2(n) + k12]11m2 L + [f2(n)gl(n) + k21]12m1 + [f2(n)g2(n) + k22]12m2 n=a fi(n)gi(n) + k11 +f1(n)g2(n) + k12 +f2(n)gl(n) + k21 +f2(n)g2(n) + k22 * The proof for the general case is precisely similar. 26 UNIVERSITY OF MISSOURI STUDIES [fi(ngi(n)n)lml +fl(n)(n)g2 lm2 +f2(n)gl(n)l12m + f2(n)g2(n)12m2 f (n)gl(n) + fln (n)g2) +f2(n)g(n) +f2(n)g2(n) T= [[fl(n)li +f2(n)12] [gi(n)mi + g(n)m2] 1=. - Lx L [fl(n) +f2(n)] [gl(n) + g2(n)] For example, the square-product of the sequences S = I, O, I, O, **t = I, O, I, O, **, is = I; 0, 0,0; 1,0, 1,0, I; 0,0,0 0,,; 0, 0 1,0, I, 1,0, 1,0; * *. If we choose m terms of this sequence, and let (2n)2 be the largest square of an even integer less than or equal to m, so that m = (2n)2 + k, o < k < 8n + 4, we get: w=L [I+3+ **+(2n-I)lI+[m-(I + -+2n —I)0' n=ol m n2 n2 = L-. + n=i m Ln- 4n2 + k - 4 -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) = Uo + U1X + U2X2 + U3X3 + * * + U.Xn + _ * v(X) = vo+ + v2x + v3x + VX + VX2 + + W(x ) V() = UOVO+ (U0ov1+U1Vo)X+ (UoV2+UV 1+U2VO)X2 + *. Furthermore, to this symbolic advantage is added the theoretical one which is contained in the following theorem, due to CesAro,* which is a generalization of Theorem B..THEOREM (J): The Cauchy-product of two Cesaro-summable series, of orders p and q, and of values s and t respectively, is itself Cesgro-summable of order at most p + q- + I, and its value is st. In certain special cases, we can slightly improve upon the results of Cesaro'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, t as has already been stated.: Similarly, the Cauchy-product of two Cesirosummable series, one of order r, the other convergent, is summable (Cr+~); 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 Cesdro-summable series of order r by an absolutely convergent series, is itself Cesaro-summable of order r. Let Sn = Ugo + u + ** + Un, tn = Vo + V1 + * * * + Vn, Wn = UoVn + UIVn-1 + *- + UnVo, yn = Wo + W1 + *.. + Wn. * Ces&ro: Bull. des Sciences math., t. XIV, 1890. t Mertens, Journal de Crelle, t. 79, p. 182. X P. 5, supra. 28 UNIVERSITY OF MISSOURI STUDIES r(r + I) r** (r I) ( + I)... (r + n - 2) Y, = y0 -- y1 n n! +Y (n- I)! r(r + I) + '"* + yn-2 2! +yn-ir'+yn, r(r ).. (r + n- I) r(r I) * * (r + n - 2) T, = to n +tl (n-)! n! (n-I)! + + tn-2 * + tn-l'r + tn, r(r + I)... (r + n-I) Tn (r, n) = t We assume:(r, n) We assume: Lsn = s, L [luol + lu1 +.+ Il = A, n==oL n=o (ro n= (r + I) T. ** (r + n) and we wish to prove: L ( n= co ( Proof: Lemma: If Yn - s't. + I) (r L,(r +) *I)... *(r +n) = t h - -e n -L n-p then rL 7-i..-r = o,,=, (r + I).. (r + n) p = 1,2, *' p. For TL r Tn I Tn-p n=o (r+ I) (r+n) (r + I) *.. (r + n) n! -n! DEFINITION OF SUM OF A DIVERGENT SERIES: T T n T n-p (r + I)..- (r +n) (r +I).. (r +n-p) =L n:8 n! (n - p)! (r + I)... (r + n-p) (n -)! (r I). (r +n) n! Tn Tn-p -= i (r+I)-.. (+n) (r+I) (r+n — p) n! (n-p)! n(n - I)... (n - p + I) (r + n) (r+n-I)' (r+n-p+ I) = t-t I = o, Now Y = UoVO + (uovl + UlVo) + (UoV2 + UlVl + U2Vo) + * * + (UOVn + UlVn-l1 + un* * + Ul + UnVo) = Uo(Vo + v + + V) + ( + v* + -V_) + Ul-1) + * - + Un-l(vo + V1) + Unvo, yn = uotn + ltn- + * * * + un-ltl + Unto. r(r + I) *... (r + n -I) Yn = uoto n + (ui+uto)r(r + I).. (r + n - 2) + (uotl + uito) (n- I)! r(r + I) + *. + (uotn-2 + ** + Un-2to) 2 —( +(uotn-l+ *. +Un-lto)r+(uotn+.* +unto), Yn = uoTn + ulTn-1 + * * + un-lT1 + unTo,.'. Y2n = uoT2n+ uTn +.ul' + ', + u2n2-1 T + uno, 30 UNIVERSITY OF MISSOURI STUDIES Let (r+I) (r+2n) -(Uo+U+ +) (r+I)..(r+n) (2n)! n! (uoT2n + ulT2n-1 +. * + u2n-lT1 + u2nTo) =~ ~ (r+ I)(r+42)... (r +2n) (2n)! Tn -(U + + ''' + Un)(rI) (rn) (r + I)... (r - n) n! R [ T2n _ n Tn T2n-_ Tn R [Ju~ I(r+I, 2n) (r+i, n) + lul (r+i, 2n) (r+i, n) 4+..+ ^ T 2n-q T +... + Uq (r + I,2n) (r+, n) T2 n-_-l Tn + ~+ [uq (r +l I, 2n) (r + I, n) + Iuni Tn, Tn 1 (r+ I, 2n) (r + i, n) [+ |+| |T, T + T +[lu 1r + I, 2n a- (r + I, 2n) T2n- To T q- f lu~+ r - I, 2n - ' -, r In) M1 + M3 + Uq+l{l {-r I 2n-g- + I n) } + *.. +un,{ (r+in) + (r+, n) ' where M1, M2 and M3 stand respectively for the expressions in the first, second and third brackets above. DEFINITION OF SUM OF A DIVERGENT SERIES 31 R < M, + M3 {IUq+ll + ** + Iunl}B, since Tm) < B for all m. (r + I, m) 2 Also M3 < {|un+l| + *. + lu2ni}B. Now as to M1, |T2n-p Tn T i T2n Tn I(r+ -I,2n) (r+i In) (r+,2n) ( In) T2n-p T2n (r + I, 2n) (r + I, 2n);.. 2(A + B) 2(A +B)if n > N;.-. R < {luol + ull + * * * + l + {Iu,+ll + ** + lu + **u * + + U2nl}B, if n > N. Now choose g so large, that lu-tll + * + lu2nl < +e > Q for all n. Moreover, luol + *. + luql < A for all q. eA + eB.'.R<- =e. Thus L rY2n n= (r + I, 2n) Similarly Y2n+l, (r + I, 2n + I) The theorem is now proved. In the case of power series, then, both the symbolic advantage and the theoretical importance of Theorems j and 5 lead 32 UNIVERSITY OF MISSOURI STUDIES naturally to the Cauchy-product. This advantage does not appear, 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 Cauchyproduct. We should observe, however, that while the squareproduct may justly replace the Cauchy definition of multiplication, 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 un+l =, then L Unl =. n=0= Un n==o Let Un+1 Un then Un = tlt2..* tn. Accordingly, whenever L t =t, then L (tl *. tn)l"n = t, provided tI > o; and the last equation may be written log = L log tl + 0log t2 + + log tn And if we finally write log tn = s,, we obtain the result that L S-F + 2 + * * * + Sn L n -00oO n whenever LSn = 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 U+ tn+ e +1. Un Let us make the further substitution n = rnpn and observe that the variables s nand rn on each side of this equation approach the same limit, provided L Pn = I. Ln= n We may accordingly replace Theorem c, which we have just obtained again, by the following theorem: THEOREM 6: If L r = r, and L P = I, n= —o n-=0o then L[n lrl + 2r2 + + nr r. If we put a further restriction on the sequence (pn we can broaden the requirement on the sequence r,. In fact, we may say: THEOREM 7: If r + r2 + ** + rn - - o = r n=*o n and IL cP I n=ao monotonically,* then * That the theorem is not true in general, when L qPn = I In= oo not monotonically, follows from the example: rn = (- I)n+1 log n, Pn = I + (- I)4+1 l n + I, = I. log n, Here L r +.~ +r =o, L p = I n= co n n= o not monotonically; L rl1 +-' + Pn= Irn =- ao n DEFINITION OF SUM OF A DIVERGENT SERIES 35 L lrl + f2 + * * + + nrn r. n=x0 n The proof of this theorem follows at once from the following theorem due to Hardy;* for a proof of which see page 85. THEOREM L: If Cn is a divergent series of positive terms, then coso + ClSl +... c so + S + S. + * + S L L,=a o+I- no +I provided that the second limit exists and either (a) cn steadily decreases, (b) c, steadily increases, subject to the condition ncn < (Co + C1 + *. + cn)K, where K is a fixed number. We shall now show that Theorem 7 is a special case of Theorem L. In the first place, since L sPn = I, n=ao it follows from Theorem c that (P1 + (P2 + * '' *+ Pn — l+- n+'"+ - = I, nIto n and accordingly, L lrl +- p2r2 +" - (nrn L L 1 (lrl ~2r2 + '*'' * * Pnrn (Pl + 2 + * + (On n=oo rP1 + (P2 + * + (Pn n (plrl + 2r2 + ' + pnrn = L,=3 c1 -+ P2 +- * + P We may now apply Theorem L directly, by identifying pn * Quarterly Journal, Vol. 38 (I907), p. 269. Hardy proves a more general theorem of which this is a special case; the first part of the general theorem has been first proved, however, by Cesaro, as Hardy himself states. See Cesaro: Bull. des Sciences math. (2), t. 13, I889, p. 51. 36 UNIVERSITY OF MISSOURI STUDIES with c,. If p,n decreases monotonically, the condition of the first part of Theorem L is fulfilled; if qpn increases monotonically, we have: <Pl 2+ + P2 + ** + (Pn > n<p\, or (pl + <P2 + * * + (Pn I >Pn n k K so that < K ( Pl 2 + P + * + n) Yn <K- - which is precisely the second requirement of Theorem L. Hence the truth of Theorem 7 is established. We can deduce an interesting consequence from Theorem 7, and say, in the language of ~ 4, Theorem 8: The natural product of two sequences, one of which is summable of order I, the other monotonically convergent, is summable of order I; and the value of the product sequence is equal to the product of the values of the two given sequences. Let sn and tn be the two given sequences, Sl + S2 4- * + Sn L S1+ 2+? + S =s, Lt =t, n=00 n n=00 monotonically. We first suppose that t = o, and form the sequence t,/t, so that L tnI n=00 t monotonically. Accordingly, by Theorem 7, tl t2 tn sl + s2 7 + * ' + Sn 7 _= oon or sltl + s2t2 + *+*' + Snt nL ----- -- = st. n=ao n DEFINITION OF SUM OF A DIVERGENT SERIES 37 If t = o, we form the sequence i + tn, so that L (I + t) = n=0oo monotonically; consequently, by Theorem 7, S1(I + t1) + S2(I + t2) + ''' + Sn(I + tn) n=n= 0 n L S _ + S2 + ' * S + St st s + s2t2 + * ** + Sntn n=Loo S + + By _-,_ _ and accordingly, Lsltl + s2t2 + * * + Sntn L = 0. n=oo n Let us now return to Theorem 6, and base upon it the following definition: Definition: The sequence shall be said to be.p-summable, and to have the value s, provided SlSl + s 22~ + + Sn Pn L --- = s. n=oo n L (Pn = I. ~=00 It is natural to ask for the relation between Vp-summability and CesAro-summability. In general it will be possible to find a sequence Vpn which will give a more general definition than that of Cesaro-summability of order I. We can however restrict the sequence Pn, so as to make the two definitions equivalent; and we may state the following theorem: THEOREM 9: If L Pn = I n=oo monotonically, then whenever either of the two definitions —psummability or Cesdro-summability of order I-gives a value to a given sequence, so will the other, and the two values will be the same. 38 UNIVERSITY OF MISSOURI STUDIES If we choose any specific sequence _ n, subject to the condition L fn = I n=0i monotonically, then it follows at once from Theorem 7 that if a sequence is summable of order I, it is also ~?-summable for the particular np. Let us now suppose, conversely, that the sequence sl, s2,.. sn,... is (-summable for p,, i. e., L s1pi S + s22 + * + Sn(pn L - - s. n=eo n This amounts to saying that the sequence (spn) is CesArosummable of order I. Let us now apply Theorem 7, making rn = snPn, and (pe = i/~n. Since L qn = I n=oo monotonically, then L jn = I n= oo monotonically, and SI= l11P1 + S2(p2(2 + * + Snn] -Sn L 51 + S2 + * * + S n=00 _ n=c n i. e., the given sequence is Ces&ro-summable of order I. If we assume that L n = I n=a= non-monotonically, then Theorem 7 may no longer apply, as is shown by the following example: qo = I si = (- I)i+1 log i = I + (- I)+ I=2, 3,*t, log i so that Sipi = O Sii = I + (- )i+l log i = I +, i = 2, 3,.. DEFINITION OF SUM OF A DIVERGENT SERIES 39 Now sl- Si+ S2 +g-' + — Sn log I log 2+ log L =L O* n=_ o n= =ao n and Lc Pn =I n =ao non-monotonically. If Theorem 7 were true, <pn non-monotonic, we should have L Si(p + S2(P2 + " + Sn o; L = o; n=oo n whereas, L sol + * + Sn.. s log I + (I + S2) +.. + (I + Sn) n-=oo n -=o-o n -n- I S2 + f + Sn =L + L S I. n=o n n=:o n' c Returning now to the monotonic p-definition, we observe that if we take n — I, we obtain Cesaro-summability of order I. Taking n =log (I +- we obtain: sl log 2 + S2 log (I + I)2 + S3 log (I + )3 + * * * - + Sn log( I +-) (6) s =L =. T 2n I/ - 3.. (2n - 1/) -L n-, (-I)i+log i - L log (I 00= o 2? i- 2 n= o 2 -4... 2? =- L1 loguln' = o 2 n= oo since L - =. An= co Un Also I 2Jr I ( -I)4+1 lo, g (2n + ) L 2 — I (-l) i1log 0 == o + 2 o. n= co 2'n + I i=1 w= oo 2n 4- I 40 UNIVERSITY OF MISSOURI STUDIES Since L log( I n monotonically, however, it follows that this definition is equivalent to Cesaro-summability of order I, or (what amounts to the same thing) equivalent to H6lder-summability of order I. If we now write SI + S2 + ' * + Sn tn.=.n +s so that ntn - (n - I)t,,- = Sn, we may repeat the process for the sequence tn, obtaining [t log 2 + t2 log (I+-})2+- + + + tn log ( -I L ---------------— v -- \ n - L SI log 2 + (sl+s2) log + +... +(S+s2+... +Sn) logn+ n=oon n + I n -i n S Il + s 2 logg + s lg + + Snogn + 2 n (7) s = L Since (6) is equivalent to the Holder-summability of Sn of order I, it follows that (7) is equivalent to H6lder-summability of tn of order I, i. e., with H6lder-summability of Sn of order 2. Let us now return to our definition of (p-summability, and repeat the process for another function t(n), where L (n) = I. n=oo DEFINITION OF SUM OF A DIVERGENT SERIES 4I Writing p=(I)Si + p,(2)S2 + '.. + P(n)Sn n we obtain L [(I)tl + (2)t2 + (n)t =LL n J (8~ *w,){,( ^)... 2(n>) (8) J Si s(I)p (I)+ 2 + +n n=Lo + S2)(2) { 2 + + 1+...+Sn( ) (n) 2 n J Now, if L (p(n) = L ~(n)= I, In=oo - n=oo then Lrn=L1 + 2 + * * * + n n=w nf=c-o n and L {(I)rl + + (2)r2 + " + (n)rn n=oo n ) (I) {t(I)+ t'(2) + + = I. 1 + &l ( ){(2)+ +s (n )}+ + (n n 2 — n J Instead of taking L s(n) = L(n) = I, n=oo 7n=oo we shall assume more generally that (9) is satisfied, and take as our definition, 42 UNIVERSITY OF MISSOURI STUDIES r ~(2) V,(n) Slq0(I ) ~(I) -{- -k-O ~ -.4. _2_ - 1 2 ( ) 4,- [r i '.~] +)^...^] L - + (2) ) ~(n___~) 1 -9 + (2) ( +) - __+ - +- *+_)[ 2 n If o(n) -t(n) i, we obtain: s SI I+ + +n +S2. + +[2 +S n=1~ 12+s2 + s 2+s2 + s +_s3_ n~co 2 3 n n L[t+~t2~+ +tnl SI +S2~+2 +S =L tl -t '"- t ' where tn= -- which is Hilder summability of order 2. If I (n) = 2on, e(n) -b- we obtain: s=L n"=o0 DEFINITION OF SUM OF A DIVERGENT SERIES 43 2n (2n- I) 2 S 2 + — I - n +( I L=X nq-I =L nsl +(n - I)S2 + + Sn,n=oo n(n + I) L 22! which is Cesaro-summable of order 2. If we put (-n - I, n = n log I +n ) we obtain: 1{log2 +log(I +) + )... log(I + )} +S2 log(I +2) +.+log (IQ+)}+ |s=L +sn{log (x+!)}| +(og0+i)-)) S -Li n --- —si {(log 2 - log I) + (log 3 - log 2) + * * + (log (n + I) - logn)l + * * * + S log- | =L n=oo n rin T I n i+I / + log I + S2 log - + - * log ] = L n^oo n which is (7). We have thus seen that the definitions of 9-summability and (Io) include some of the specific definitions which we have already discussed. One might naturally ask, however, whether these general definitions themselves may be of any use. One use immediately presents itself, as can be seen in the following example. 44 UNIVERSITY OF MISSOURI STUDIES It is desired to know whether the series given by Si = -( / I\ i = odd ilog (l+ =, i = even is summable* according to Cesaro's definition; and if so, its value is required. To determine this directly from Cesaro's definition requires some manipulation. If we choose, however, i = log I + I, we obtain sl(Pl + S2(P2 + + Sn<sn _ L I +0+ I+o+.-. +o or I n=oo n 1n=00 n n n - or -+ I 2 2 n=0_ n And since L (Rn = I monotonically, it follows that 3L + S2 + * S.n L = 2. Wn=oo n This example leads us to formulate the following proposition, which is of practical importance: THEOREM IO: To test a given sequence for Cesaro-summability of order I, any convenient (pn may be chosen, provided L SPn = I n=I monotonically. Similarly we may sometimes simplify our calculations in testing for Cesiro-summability of order 2, if we can find a suitable (pn and i,. * This example has been already considered from another standpoint. See p. 22. DEFINITION OF SUM OF A DIVERGENT SERIES 45 We might now proceed to generalize to p-functions, and show that the resulting generalizations would include all of Cesaro's and Holder's definitions. And from what has preceded, it is easily seen that if we take all the p-functions equal to unity, we shall obtain all of Holder's forms; while by a suitable choice of these p-functions, all of the Ces&ro-forms might also be obtained. 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, n=oo (ii) L [ai(n) + a2(n) + * + an(n)] = I (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 Ceshrosummability of order r, we shall show that the sum of the coefficients of the numerator, divided by the denominator, is identically equal to unity. For this purpose we write: (I - X)-(r+l) = (I + X + X2 + X3 * * + Xn + * * )(i - X)-r Equating the coefficients of xn on each side of this identity, we obtain: (r + I)(r + 2) (r + n) r(r+ I) —! - - I + r +-... r(r + I) *- (r + n - I) + n! * We exclude also definition (Io). 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.! f *"+ 2! +r+I (r + I)(r + 2) *. (r+ n) n! 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 considered. Suppose now that hi, h2,.. hn are the coefficients of Holder's definition of order p, so that L [hisl + h2s2 + + hs] = s. If we assume that hi + h2 + * + hn -- I for order p, we obtain for order p + I, putting tl + t2 + * + tn Sn n L [hit + h2 tl + t2+ + h tl + "+ tn] 2 r. +fl, i h2 Jh h =L [tl i+22+. + n +.. + J -n and the sum of the coefficients becomes [h + 2 +'+ ]+[ 2+ + n -- +- +n 2 n 2 n n _ hi + h2 + * * * + hn =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 L Un I, ra==ao 48 UNIVERSITY OF MISSOURI STUDIES where n n n log- + log-+ * * + log 1 I 2 n —I n n n u = = ---— l-og-* -..... --. nn = \ (I 2 n- I If n n n I n =-.-. —. --- nn-l I 2 n-I (n- I)! then n ( n Hence L vln = L Vn+l e _=oo n=o vn Accordingly, L un =L log v1nl = I. n=oo n=oo Finally since we have assumed in the r-definition that L ((n) = I, =oo 00 it follows that L (I) + Sp(2) +... + ~(n) =00o n by Theorem c. Thus it is seen that all* of these definitions have properties (i) to (iii) in common. We can accordingly generalize our notion of summability by stating a definition in terms of these properties themselves. Definition: A series shall be said to be A-evaluable,t and to have the sum s whenever the following conditions are fulfilled: * Except definition (Io) t We shall hereafter use the term evaluable in the case of definitions in terms of properties of general functional coefficients of the si; the word summable we shall retain for concrete definitions with specific coefficients. DEFINITION OF SUM OF A DIVERGENT SERIES 49 (i) ai(n) = o, for fixed i, (ii) L [a(n) + a2(n) + ** + an(n)] = I, (A) n=( iii) a(n) > o, (iv) L [al(n)sl + a2(n)s2 + +. a,(n)sn = s. n=mo We shall now justify this definition by proving the following theorem: THEOREM I I: If a series is convergent then it is A-evaluable.* By (iv) we may write: [al(n) + a2(n) + * * * + an(n)] + rn _ I, (v) n Lrn = o. 3= —oo Now, by (v), I ai(n)sl + a2(n)s2 + - * + an(n)s, - s -I {a(n)si + a2(n)s2 + * * + an(n)sn} - (al(n) + a2(n) +. + a,(n) + rn)s I < | al(n) (sl - s) + a2(n)(s2 - s) + * * * + ap(n)(sp - s) l + I ap+i(n)(sp+i - s) + * * + a,(n)(s, - s) I + | rs|. Since the series is convergent, we can choose i so large that si - sl < r, i > p. Let I be the largest of the numbers I si - s 1, for i = I, 2, *.. p. We have, then, I al(n)sl + a2(n)s2 + * * + a,(n)s, - s 7 I{a(n) i si - s | +' * * + a(n) I sp - s } * Theorem I I obtains if condition (iii) is replaced by the broader condition: I al(n) I + I a2(n) I + '- + I an(n) i < K. 50 UNIVERSITY OF MISSOURI STUDIES + {ap+l(n)lsp+l - sl + -. + an(n)lsn - s } + rnsl < {al(n)+ * -+a+(n)}l++ {ap+l(n)+ * * +a(n))} 7+Irnsl < l + n + rns I, n > N* e e e e e 3 3 3 by31'), 3 - e. Hence L [al(n)sl + a2(n)s2 + + an(n)s,] = s. n=oo Our definition of A-evaluability is now justified. The question naturally suggests itself as to whether for a sequence (s,) which diverges to + oo, L ai(n)si = + o. nawoo i=1 The answer, which is in the affirmative, is embodied in the following theorem: THEOREM I IA: If L n = + 00, n=oo and conditions (i), (ii), (iii) are satisfied, then L ai(n)s = + oo. By hypothesis, Sn > N, n > m. Hence n m n,n = ai(n)si = ai(n)si + E ai(n)si *=1 i=1 m+1 m n > ai(n)si + NV ai(n). __________ =1 i=mS+ * By (i), [al(n) + * + ap(n)] < 8, n > N, p having been chosen first, and then held fast. By (iii), [ap+l(n) + * * + an(n)l < [ai(n) + * * + an(n)l < I by (ii). DEFINITION OF SUM OF A DIVERGENT SERIES 51 Since mn n ~ L ai(n)si + N E aI(n) = N, n= —o L= i=m+ 1 it follows that Minimum L c, >_ N; n=00 and since N is an arbitrary number, L-n +. 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, S o+ 52 ++, S+s+ -S I~~~~2 n n () = Lo log n n= I I This formula is of interest to us, since it affords an example of a definition which is broader than CesAro-summability of order I, and yet perhaps not so general as that of order 2. For since i/n steadily decreases, it follows from Theorem 8 that formula (ii) gives a value to all series that are CesAro-summable of order I, and that these values are the same for both definitions. That (ii) 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 Un L - o; n —oo however we obtain from (iI), for the corresponding sequence, Sn = (- I)f+ln, 52 UNIVERSITY OF MISSOURI STUDIES L [o I -,I + I - I.." — I] [ I or o =0 log n = Elog n 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 (i ) is L - = o.* n=o n log n This is not, however, a necessary condition for summability of order 2t-so that we might find a series for which Un n=0 n log n which is nevertheless summable of order 2. We have seen that the A-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 Borelmean-definition, we shall now generalize Theorem I I, as well as the definition which we have based upon it. Replacing ai(n) 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: I I - Si + 2+ * * * + S+i S in+ Sl +2S2+ * * + - Sn 1 *0 ^ o I ~...^ i+-..+ J Un Sn Sn-1 L =L -Sn-= 0. n=oo n log nn=oo n logn t A necessary condition for summability of order 2 is Un L - =O. n= — n2 See p. IO. DEFINITION OF SUM OF A DIVERGENT SERIES 53 (i) L ai(a) = o for fixed i, a-' oo (ii) L [al(a) + a2(a) +..* + an(a) I, (iii) ai(a) > o, (iv) L sn = s, n=oo may be deduced the result: L L [al(a)sl + a2(a)s2 + - + an(a)sn = s. a= co n0= O We shall first show that L [al(a)sl + a2(a)s2 +.. + an(a)sn, n=oo exists for every definite a. Taking a definite value of a, [an(a)s o + an+l(a)sn+l +... + an+p(a)sn+pl < an(a)lSnl + -** + an+p(a)lSn+pi <A(- * A by (ii) ((n > N, any p)) - e. Hence oo E an(a)Sn n=l1 converges for every value of a. Since 00 E an(a)Sn =l1 has a sense, we may write: 00 00 00 Ean(a)sn- - S an(a)sn- an (a) s by (ii) n=l n=1 n-t 54 UNIVERSITY OF MISSOURI STUDIES oo rn-1 co n-) an(a)(Sn - s) 2 a,(a)(s- s)- a,(a)(sn-s n=l n=l n=m m-1 < HE an(a) + e, =l1 since Isn - sl < e, n > m, and Is, - sI < H, n < m by (iv). Since, however, m-1 L a(a) = 0 a oo n=1 by (i), it follows that: oo rn-l Maximum L E an(a)sn-S <e e+Maximum L H E an(a) =e. a-=oo n= a-oo -=l Since e is arbitrarily small, the maximum limit on the left must be zero, and therefore the actual limit is zero, i. e., 00 L Zan(oa)sn=s. oo a=oo n=-l It is readily seen that Borel's mean-definition satisfies conditions (i) to (iii) of Theorem 12. For we have, in satisfaction of condition (i), an L = o; a=oo e that condition (ii) is satisfied follows since el a2 an + f-., +... L a —'J eI and finally, since an"/ea > 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 12 as a basis. It turns out, however, that we may generalize Theorem 12 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) L ai(a, n) = o, n=o00 (ii) L ai(a, n) - I, -n=oo t=O (iii) ai(a, n) > o. If now these conditions are fulfilled for a fixed value of a, and if L n = S, n=oo it follows from Theorem II, that L ai(a, n)si = s. n=oo i=O Since this limit exists for every value of a, under our hypothesis, we may write: (iv) L L E ai(a, n)si = s, a=ao n=oo i=O 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 Uo + Ul + U2 + + Un + * has the value s, then the series uI + u2 + * + Un + * * must have the value s - u0; * The same is true, of course, for the A-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 Sn = uo+ + ul +-.* + un, when the sequence so, sl, s2, s..., has the value s, then the sequence si - Us, s2 - UO, *. - u *,.. has the value s - Uo. If we assume, for the moment, that whenever either one of the two sequences SO, Si, S2, *'' Sn, *'* Sl, S2, * '' Snn, *. has the value s, the other does also; then we shall satisfy our third requirement if we prove that whenever s^, S2, S,... Sn,.. has the value s, then s1 - Uo, S2 - UO, S3 - u0o,.. -, *. has the value s - uo. Now this it is easy to prove. For we have by iv, p. 55, n n L L E ai(a, n)(si-uo)=- L L E ai(a, n)si-uo=s-uo a=oo n=oo i=O a=oo -n=oo i=O by (ii), p. 55. It remains then to consider under what restrictions we can justify our assumption that the two sequences So, S1, S2, '' Sn, * S1, S2, *. Sn * *. 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: r o a a2 an s(a) = So + s5 I + S2 + * * * + Sn - + * * *, n —l a a S,(a) = s+S + S2 +. + Sn ( )+ ' a a2 an — S'(,a) - s(a) = U U + U2 I + - + * * * + Un, ( - I) + * * * Borel's definition being = L s (a.) DEFINITION OF SUM OF A DIVERGENT SERIES 57 If we assume* that L s(a) = co, a=00 we have an indeterminate form, so that s(a)1 S'(a) 1L ea == L a Y a= O ea a= oo e or L s'(a) - s(a) a=00- a ~0, which may be written, L L e [U+2a a2 a l Ie-a u+u2 1 ++.+ - + +un+1:o It is accordingly suggested that we assume, in general, (v) L L [ao(a, n)ul + al(a, n)u2 +... + an(a, n)Un+l] = o. a=ao n=o As a matter of fact, this condition is sufficient,t for, from (iv) (iv) L L [ao(a, n)so + al(a, n)sl +. * + a,(a, n)s] = s and a=ao =oo0 adding (iv) and (v) we obtain L L [ao(a, n)sl + ai(a, n)s2 +... + an,(a, n)sn+ll = s a=ao n=O which proves that when the sequence so, si, *.. sn,... is evaluable to s, so is the sequence s1,, 2 Sn, * *. By subtracting (v) from the last limit we show in the same way that when the sequence, s,... s,, n... is evaluable to s, so is the sequence SO, S1, S2, 2 sn, *... 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 general 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 ai(n, a). It would be desirable to have on ai(n, a) further restrictions, sufficient to cause (v) to hold for all sequences. 58 UNIVERSITY OF MISSOURI STUDIES Definition: A series shall be said to be B-evaluable and to have the sum s whenever the following conditions are fulfilled: (i) L ai(a, n) = o, ~-=00 n (ii) L A ai(a, n) - I, n=oo i=O B (iii) a(oa, n) > o, n (iv) L L Z ai(a, n)s = s, a=ao n=oo i=0 (v) L L E a(a, n)ui+ = o, k = I, 2, p. a-ao n=oo t=O 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 L Sn = s, -t=00 then e-u(r)dr = s, where r r2 rn u(r) = Uo + ul + u2 - + * * + un + *. 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. IOI; Hardy proved the second proposition by an example: Quarterly Journal, Vol. 35 (I903), 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. t This definition does satisfy requirement (iii), as Borel proves, T and it obviously satisfies requirements (ii), (iv) and (v). Furthermore, Borel makes the statements 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: Un =., n = i2, Un = o, n not a square. In fact the series in question: - I + O +-oI+ -+ O + O + O + O - - + is convergent, while s e-ru(r)jdr is divergent. Thus, since absolute summability fails to satisfy * Loc. cit., p. 99. t See p. 14. t Loc. cit., p. Ioo. ~ Hardy, loc. cit. 60 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 fundamental requirements, and that it includes as special cases all of the proposed definitions of summability which satisfy those requirements. Our definition of B-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) L [al(n)sl + a2(n)s2 + + an(n)s,] = s, 0= oo (b) the fundamental requirements of page 2 are satisfied. Definition 2: An abstractly-evaluable series of functions of a variable shall be called uniformly evaluable, if: L [al(n)sl(x) + a2(n)s2(x) + +... an(n)s,(x)l 1C1 1)=00 (c = L f(x, 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. t For f(x, n) = al(n)sl(x) +... + a,(n)s,(x) is a continuous function of x; and since Lf(x, n) = s(x) n=oo 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 6i THEOREM I3A: A suificient 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 convergence. * THEOREM I3B: A necessary and sufficient condition that an abstractly-evaluable series of continuous functions define a continuous function is that f(x, n) have Arzela's quasi-uniform convergence. t THEOREM 14: A uniformly evaluable series of continuous functions may be integrated term by term. We wish to prove in this case that L [ai(n)sl(x) + a2(n)s2(x) + * + an(n)s,(x)ldx ~ n=oo rb =L [al(n)sl(x) + a2(n)s2(x) + + an(n)sn(x)]dx n=- 00 0 or Lf(x, n)dx = L f (x, n)dx, -n= oo n= oo a 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 ai(n), bi(n) render the series abstractlyevaluable at Xo, to the values s and t respectively; then, if the evaluability of each of the definitions is uniform in the interval, then s = t. Letting f(x, n) = ai(n)si(x), i=O and n g(x, n) = bi(n)si(x), i=o * Dini: Fundamenti per la teoretica delle Funzioni di variabili reali. Pise, 1878, p. 103. t Arzela: Memoires de Bologne, I899. 62 UNIVERSITY OF MISSOURI STUDIES and remembering that since the series is convergent, x = xo, it is true that Lf(x, n) = Lg(x, n), x = xo, n=00 X = 00 we have from the uniformity, L Lf(x, n) = Lf(xo, n) = s, x=xo n=oo ~=oo L L L g(x, n) = L g(xo, n) =,t x=XOn=oo ~= 0o and hence s = t. We may obviously state the preceding theorem in the following more general manner: THEOREM I5A: If a series of functions continuous on an assemblage (E) is convergent at all points of (E), except possibly at x = Xo, which is a limit point of (E); and if two sets of functions ai(n), bi(n) render the series abstractly-evaluable at xo, to the values s and t respectively; then, if the evaluability of each of the definitions is uniform on (E), 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.t Let us seek for a continuous solution of the integral equation, fib u(x) = f(x) + K(x, )u(t)d, ua where K(x, y) is continuous,$ Ha <x <b\ a < y <b and f(x) is continuous, a < x < b. Following the method of Volterra, we shall form the iterated functions: Kl(x, y) = K(x, y), (12) b (I2) RK,(x, y) = K,(x, K )K_1(-, y)d~. Then Ki(x, y) = K(x, 1)K(,, ~2)... K(_l, y)d-il * * d and Ki+j(x, y) = Kf(x, c)Kj(%, y)d. * Our result is more general if we restrict ourselves to Volterra's method; a much more general result has been obtained by Fredholm by means of a different method. See Acta Math., Vol. 27 (1903), p. 365. t Rendiconti, Accademie dei Lincei, series 5, Vol. 5, I896. t The theorem can be proved with much broader restrictions on K(x, y). 63 64 UNIVERSITY OF MISSOURI STUDIES If we first put i = I, i + j = m in this formula, and then put j = I, i + j = m, we obtain:* (I3a) Km(x, y) = K(, )Km_(i, y)d~, (I3b) Km(x, y) = Kf ml(x, -)K1(, y)d. Volterra now proves that if the series Ki(x, y) + * * + K,(x, y) + *.. converges uniformly in s, then the integral equation has one and only one continuous solution. We shall prove, more generally, the following theorem: THEOREM 16: If the series KI(x, y) +. * + Kn(x, y) + * * is uniformly evaluable in the abstract sense, then the integral equation has one and only one continuous solution. o00 Since E Ki(x, y) is evaluable, i=l - k(j, y) = K1(%, y) + K2(J7, y) +... + Kn(, y) + *.., and by our fundamental requirement (v), p. 2, - k(~, y)Ki(x, K) = Ki(x, K)K1(j, y) + Ki(x, )K2(, y) + + Ki(x, )K(, y) +... Moreover, the last series is uniformlyt evaluable. Hence we may integrate term by term, by Theorem 14, obtaining - fK(x, i)k(, y)d = K(x, )K1(x, y)d a ba +f K(x, i)K2(Q, y)d + ** +J Kl(x, )KI,(, y)d +*** = K2(x,, y) + Kn+(x, y) +... +K+ y) * The first of these two formulae is the same as the definition of Km(x, y). t The uniform evaluability can be established in precisely the same way as in the case of convergence. DEFINITION OF SUM OF A DIVERGENT SERIES 65 by (I3a). The series last considered has for its value, - k(x, y) - Kl(x, y) so that br7 f K(x, t)k(I, y)d~ = K(x, y) + k(x, y). By using (I3b) in a similar fashion, j k(x, )K(%, y)di = K(x, y) + k(x, y). The rest of the proof is the same as that given by Volterra,* who obtains as the unique continuous solution: (14) u(x) =f(x) - k(x, ~)f( )d,. It is not difficult to construct an example for which the series Kl(x, y) +.. + Kn(x, y) + *. does not converge but is, for example, Cesaro-summable of order I. Let us look for a continuous solution of the integral equation: 2 [f u(x) = I +- ( sin (x - y)u(y)dy. Here 2 2 Kl(x, y) = -sin (x - y), K2(x, y) = - cos (x - y), 7r 7r 2 2 K3(x, y) = - sin (x - y), K4(x, y) = -cos (x - y), 7C 7r and so on, so that the series becomes - k(x, y) = Ki(x, y) sin (x _y) COS = -sin(x - -cos (x-y) (I - + I - +* *), which is not convergent. Its summable value (CI) is, however, - k(x, y) =-[sin (x - y) - cos (x - y)] * Volterra, loc. cit. 66 UNIVERSITY OF MISSOURI STUDIES so that our solution will be: I 'r u(x) = I + - [sin (x - y) - cos (x - y)]dy. irJo An interesting application of Cesaro-summability of order I has been given by L. Fej6r.* It is well-known that if a function f(x) satisfies Dirichlet's conditions, it may be developed into a convergent Fourier series. Fejer has shown that if f(x) is finite and integrablet and of period 2 7, then the Fourier development corresponding to f(x) will be Cesaro-summable of order I to the value [f(x + o) + f(x- o)] at all points at which the function is continuous or has a finite jump. A similar result has been obtained for the development in terms of Bessel functions by C. N. Moore.+ We proceed to the consideration of a similar theorem in the case of the development of a function in terms of power series. If we write: h2, n-1 (I5 S = f(a) + hf'(a) + f "(a) + + (n - - Rn = f(a + h) - sn,, then Taylor's Series with a remainder may be written f(a + h) = Sn + R,, where it is found, on the assumption that f'(x), * * f(n>(x) exist, in the interval (a, a + h), that ~ hn (16) Rn =-fn(a + Oh), o<0 < I. * Math. Annalen, Bd. 58, I904, p. 51. tf(x) may become infinite at a finite number of points. t Transactions, Am. Math. Soc., Vol. Io (I909), p. 391. ~ This is Lagrange's form for the remainder. See Goursat-Hedrick, loc. cit., p. 90. DEFINITION OF SUM OF A DIVERGENT SERIES 67 From (15) it is obvious that h2 hn (17) f(a + h) = f(a) + hf'(a) + 2f" (a)+ + f- (a) + if and only if L Rn =o. n=00 If it should turn out that L R, = k * o, n=o then it follows that the series of the right member of (17) cannot represent f(a + h). But if L Rn 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 (I6) -, = f(a + h) - Z Ri = f(a + ) -R, I n - n i=1 As before, we consider three possibilities. If L Rn = o, n=oo then L ~ s = f(a + h); if LR = k o, L ESi f(a + h); n=00o 00 ln=oo n and if L Rn does not exist, L - si does not exist. w=oo n=oo n i=1 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 Rn. We have: h h2 nf(a) + (n - I)f'(a) +- (n-2)f"(a) - +. (19) hn-2 hn-1 - +2f-(n-2)() ( ) + n-l(a) ( I) -t2Jn w(n - 2)! (n - For fixed a and h we let the difference I hp f(a + h) - Si = P = Rn, n i —1 and we consider the auxiliary function r(x) = ( ) - () + ( - nfI) (a + h)-x)-h-(x) (a + h - x) + 2 (a + h - x) n-2 + (n - 2) f"(x) + f... + - 2 f(-(x) 2! (n - 2)! (a + h - ) (-l ) (X) + (a + h - X)PnP]} (n- x)! f Since p(a) = p(a + h) = o, it follows that p'(a + Oh) = o, o < 0 < I. But n(x'() =- [nfx) + n - I) (a - )f() +- (n-2) ( f"'(x)+.+ ( - ] + - ( n - 2 ) (ah-x)2 ) (a + h-x)2(^ +... +(aa + h —x)n-lnp + n - [ ( -~ ffx )+... + fn-X) +(a (_ + h - X)2' '(+ h- xP] + (n - i)f'(x) + (n - 2) f + + (n- 3 1(x -( ) ha+ -+ x)P -, (ha h -X)-lfn(x) _ (a +- h-x)r lnP (n — I)! DEFINITION- OF SUM OF A DIVERGENT SERIES 69 Since qo'(a+ Oh) = o, - X kX2~ + P n= I {f'() + I.f"(.) + "'() + * Xn-2 Xn"-1 +(n-2)f (n) + (n - I)!( where ~ = a + Oh, a + h - h( - ) = X, o < 0 < I. If we choose p = i, we obtain: h _ X f,,( X2... R = hP = f'(- ) f + f"( )+ f"() + (20) X"-2 Xn-1 4- -- --- n"-1!'?- ffn(~)} + (n - 2)! (n -(- I)! ' If now -, s, =f(a+-h)-Rn, fn i=1 then L - ssi=f(a+ h) n9=oo / i=1 if and only if LRn =o. n=00 We have thus proved: THEOREM I 7: If the first n derivatives of f(x) exist in the interval (a, a + h), then h h2 hn f(a + h) = f(a) + f'(a) +- f"(a) +- + fn(a) +, where the infinite series is Cesaro-summable of order i, provided LRn 0 n=00 where 70 UNIVERSITY OF MISSOURI STUDIES hC_ f X2 Rn fl() ++ ll ) ( t-') + * Xn-2 Xn-1 + - fn- )+ fn( + ) P +(n - 2)! "- (n - = a + Oh, X = a + h -, o < 0 <I. Turning now to the c-definition,* n E CiSi =L i i=1 we may obtain a form for the remainder similar to (20). We shall put n Z (i = (pin j=i and obtain n 2 si~pi 2 i=l ' +f(a) 1 hf'(a)qP2n +. f"(a) a3, +nt+(in- 1) We now define P by the relation: 5^1 hP f(a h) - =-P = R (Pin P * This definition is the same as that on p. 37, since n L i= = n=oo n because L pn = I. n=oo DEFINITION OF SUM OF A DIVERGENT SERIES 7I and we construct the function: I,l, f(a - h - x)-f < (x) = - f lnf(a + h)-[ <lnf(x) + ct2n I!f'(x) (a + h- )2 ) (a + h- x)_-l + V~ — a f"(x) --- +... * +,Pnn f — l(x) (a + h - x)P ] p Since p(a) = (a+h)=o, we must have sp'(~)=o, ==a+ Oh, o < < i. But (ah-x) (a~h-x)f f)=l f(+ (a+h- X)) n-1,,,(x) + * * * + ( -) fn(x) -(a + h- x)P-lPnP } so that: I X X2 f,,,( p { lf () + fP2. f () + (P3^ f"'() + + o (n -- I) f"(O) and accordingly, if p = I, (21) Rn -- pf'() + X2,() + f( +. \xn-1 1 ~ +p (n - I)!f () We now turn our attention to the form of Rn for the Adefinition. We set E a(n) = ai j=i and obtain: hn-l ai(n)si = f(a)al, + hf'(a)a2, + ** + ( - f (a)ann. 72 UNIVERSITY OF MISSOURI STUDIES We define P by the relation alnf(a + h) - a iSi = hP = Rn,* i=l and we form: o(x) = a,,f(a + h) - ainf(x) + a2,! f'(x) (a + h -x)2f (ah - x) n- 1 -f 2! (n-i)! + (a+h-x)P] V Since (p(a) = p(a + h) = o, we have for x = a + Oh = h, = (X) = -a(n)f'() + a2(n) a+h- xf "(x)+ + a,(n) (a + h - x) n-1 +a ) (n-)! f(x), so that if, as before, h(I - 0) = X,;k2 P = aj(n)f'( ) + a2(n) f"( ) + a3(n) f"'(S) +... + an(n) ( n I) ( and accordingly, X X2 Rn- = h al(n)f'( ) + a2(n) f" () + a3(n) 2. f"'() + ** + an (n) (n ) ) ] We may now state our result so as to include Theorem 17 as a special case. THEOREM 8: If the first n derivatives off(x) exist in the interval (a, a + h), then * We previously assumed the form (hp/p)P and found p = I most convenient; we here choose p = I at the outset. DEFINITION OF SUM OF A DIVERGENT SERIES 73 h hh2 hn f(a + h) = f(a) + -f'(a) + -f"(a) + + X ~fn(a) +.., the infinite series being A-evaluable, provided LRn = o, n-=oo where Rn = h [a(n)f'(a) +a2(n) f"() + + a,(n (- I!fn)] =a+ Oh, X =a + h-, o<0<I. We proceed now to the proof of a theorem which will again illustrate the possibility of obtaining results from very general definitions. Any specific definition for the value of a sequence shall be briefly designated as a D-definition, if it satisfies the following requirements: (I) the definition gives the value s whenever L Sn = s, n=oo (2) the definition gives oo whenever L sn = oo. n=00 It will be observed that every definition we have considered, either of summability or of evaluability (except* Borel's absolute summability), is a D-definition. t It is known that if a series converges for every rearrangement of its terms, it is absolutely convergent. We now prove the following more general theorem: THEOREM I9: If corresponding to every arrangement (r) of the terms of a series, there exists a D-definition (Dr) 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 (I) 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 requirement, some of which are included in Theorem I Ia. 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 D-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 UI + U2 + U3 +. - V - V2 - V3 - 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 u-series, is divergent. We can accordingly choose ki terms from the u-series so that EUi > VI + I, i=l then the next kz terms of the u-series so that E Ui> V2 + I, i=ki+l and so on. Now consider the arrangement kJ k2 Ui - Vi + Ui - V2 + = i=kl + 1 The sum of the first 2n terms is greater than n; and the sum of the first (2n + I) terms is greater than n + a positive term. Hence the series diverges to oo for this arrangement, and accordingly the corresponding D-definition gives it the value 0o, 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 introduce, 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 evaluable when all the terms are made positive, it must be convergent; for otherwise it would diverge to oo, 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 I9, absolutely 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 ul + u2 +- u + * be summable (Cr) is L = o.* n=oo n THEOREM (3): A reducible averageable sequence with a finite number of strong limit points is Cestro-summable of order I. This is a sufficient condition for summability (C1). 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 (C1). Let the series be ul+u2+u3+**-, and s.n=u+u2+* -+un. In case (a) we have s2m-1 > s2m+l > s2; S2m-2 < S2m < s. In case (b) we have s2m-1 < s2m+i < A; 52m-2 > 52m > A. Hence in either case, L s2m+1 exists = 11; Ls2m exists = 12. By Theorem 3, -"=oo m 0oo therefore, the series is summable (C1). As examples, we may take: (i)22 - 3 + - +..., *See p. II. 76 DEFINITION OF SUM OF A DIVERGENT SERIES 77 (ii) i - 1 + (~)2 - ()3 + (4)4.... (iii) I - I.I + I.II - I.III + I.IIII -. Examples (i) and (ii) illustrate case (a); (iii) illustrates case (b). THEOREM 21: Let n n Sn = Ui, Sn = Si i=1 /n i=1 00oo then the series u ui is summable (Ci) if either (a) Sn< Sn+l < A, i=l n> Nor (b) Sn > s,+l > B, n > N. For 1 =I[~S7 1+ S2+ +'' Sn-l1 I= Sn-]. Sn - Sn-1 = [SSn - Sn-11 -Now by (a), Sn - Sn_- > o, and Sn < A. Hence L Sn exists. n=-00 Similarly for case (b). 00 THEOREM 22: Let a series ui satisfy the conditions i=1 (a) the series is summable (C1), (b) \Sn\ = U1 + U2 + - Un < A, and let a set of positive constants ei be given such that either (c) ei > ei+l or (d) e, < ei+l < A, i > N; then the series elu1 + e2u2 + * * is summable (C1). By (c), L en = k, and en > k. n=o00 00 If k = o, ~ euii is convergent by a well-known theorem,* and i=l hence is summable (C1). If k = o, let n = en - k > o. Then 00 5n ~n+1 >: o, and L r n = o. Accordingly* the series biui n=oo =1 00 is convergent, and hence summable (C1). But E kui is sumil mable (C1) by (a); so that * See Goursat-Hedrick, Mathematical Analysis, p. 349, ~ i66. 78 UNIVERSITY OF MISSOURI STUDIES 00 00 E (6i + k)ui = eiui i=1 i=1 is summable (C1). Similarly for case (d). If in the preceding theorem we put 00 ui = I - I + I - I -.., i=1 we obtain: Corollary I: If the terms of an alternating series monotonically decrease in absolute value, the series is summable (C1). 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 (C1). Since, if Isin < A, then Iun = Is, - Sn,-1 < 2A, this corollary includes Theorem 20, case b, as a special case. Before proceeding to sufficient conditions for Ceshro-summability of order higher than the first, we shall prove the following theorem,* which we shall soon need. THEOREM 23: If V = V1 - V2 + V3 - V4 + * * is an alternating series whose terms decrease monotonically in absolute value, then the Cauchy-product of V by the series I - I + I - I + * is summable (C2). By Theorem 20, the series V is summable (C1); hence the product is, by Theorem (j), surely summable (C3). We wish to show that it is summable (C2). (V-2+3- )( - V2 + - + )( - I - I...) = v - (V1 + v2) + (VI + v2 + ) - **. The sequence corresponding to this product series is: (a) v1; - v2; v1 + v3; - (v2 + v.); (v)1+ Vs + v5); * More generally, if U and V are two alternating series whose terms decrease 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 Cesaro's first mean is: V1- V2 (VI - V2) + (V1 + V3) 2(V1-V2) + (V3-V4) ( ) v,; -;.... 2 3 4 Let us write the odd and the even elements of this sequence: n(vl - v2) +(n - )(v3 - v4) +. + (v2- )( - V2n) t2n=2 [n(vl -V2) + (n - I)(V3 - V4) + * t+ (V2n- - V2n)] + (Vl + V3 + * + V2n+l) t2n+l = =- 2n + I Now (vl - V2) + (V3 -4)+ + + (V2n-l - 2n) + is convergent; for if Sn denotes the sum of the first n terms of this series, we have Sn-1 < Sn < VI, since Vn+l < v,. Since L Sn exists, n=oo sl S+ S2 +* "' * Sn 1=00 also exists, i. e., n(vl - V2) + (n - I)(V3- V4) + + (v2n.-l - 2n) - L ----- = L2t2n -=00o n In=oo exists. Furthermore, since Lvn exists (owing to the relation 9b= 00 O < Vn+1 I Vn), L v2n+l = 1, -=00 and hence L 1 v + V - + + 2n+l -= =. Thus 2n vl + V3 + * " + V2n+l n Lt2n+l = L t2n - + + +- + =nW e =a o on t 91=00 I 2en +- I' and each of these limits exists. 80 UNIVERSITY OF MISSOURI STUDIES Thus by Theorem 3 the sequence fi, having two and only two limits of equal weight, is summable (C1). Hence the sequence (a) is summable (C2); which we wished to prove. If, in addition to the hypotheses of the preceding theorem, L n = o, n=00 then L +V3 +'1 + 3 + + 2n+l L = 2 = 0, n=oo 2n + I and L t2n+l = L t2n. n=o0 n=oo00 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 (C1). We now return to sufficient conditions for summability. THEOREM 24: Let U1 - u2 + u3 - u4 +.. be an alternating series, ui > o, and AuiP >2 o; then (a) if A2ui < o, the series is summable (C2); and (b) if in addition L Aun = o, the series is n=o00 summable (C1). Case (a). Consider the series: u1 - Au1 + A2 - u Au3 + * Since Aui > o, this is an alternating series, and since A2u, < o, either A2ui = Au+l - Aui < o, or the terms decrease monotonically. Hence by Theorem 23 the Cauchy product (U1 - AU1 + Au2 - au3 + **-)(I - I + I - I **- ) which is = U - (u1 + AU1) + (U1 + AU1 + au2) - = U1 - U2 + U3 - U4 + * is summable (C2). Case (b). Here the series u1 - Au1 + Au2 - AU3 + *. * Bromwich, Infinite Series, p. 350, ex. 9. This is a special case of Theorem 27, below. t auj = Ui+1 - i; A"nU = AC(An-1U). DEFINITION OF SUM OF A DIVERGENT SERIES 81 satisfies the hypothesis of Theorem M, since the terms decrease monotonically to zero. Hence the product series U1 - u2 + U3 - U4 + -. is summable (C1). Thus, for example, the series i - (I +) +(I + )2 ) - I - log 2 + log 3 - * are summable (Ci); while the series I - 2 +3 - 4 + 22 - I 32 + I 2 + I I - + +.-. 2 3 4 are summable (C2). THEOREM 25: If in the series u1 - u2 + u3s *, ui > o, AkUi > o, Ak+ui < 0, then the series is summable (Ck+2); if, in addition, L Ak n = o, n= 00 then the series is summable (Ck+l). Let I - I + I - = A, dk = AkUl - AkU2 + AkU3 - * * *, do = U1 - U2 + U3- * Then do = A(ui - di) di = A(Au - d2) dk-l = A(Ak-ll - dk) Substituting the value of di in the expression for do, do = Aul - A2(Al - d2). Substituting for d2, d3, and so on, in turn, 82 UNIVERSITY OF MISSOURI STUDIES do = Au1 - A2Au1 + A3A2U -... =4- AlAk-ul =F A1dk. Now dk is an alternating series whose terms decrease monotonically in absolute value. Hence dk is summable C1, and Akdk is summable* (Ck+2). Since do -= Akdk consists of a finite number of terms each of which is summable (Ck), or of lower order; it follows that do is summable (Ck+2), and the first part of our theorem is proved. If we now further assume L AkUn = o, n=oo it is seen that dk is convergent, and Akdk is summable Ck+l. It follows, accordingly, that do is summable Ck+i. * It can readily be proved that Ak is summable (Ck). ~ 9. THEOREMS ON LIMITS The object of this section is to emphasize the value, from a practical point of view, of Theorem 1, which we restate for the sake of convenience: THEOREM II: If (I) L ai(n) =o, for all i, n=Qo n (2) L ai(n)=, n=-oo z=l (3) either ai(n) ~ o, or lai(n)l < k,* (4) Ls, = s, or + 0o,t n=oo then n L ai(n)s = s, or + oo respectively. We have pointed outs that many of the definitions of summability 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 oo, then LA n A n+l- An L = L =0o Bn ~q=o Bn+1 - Bn provided that the second limit exists, or is + oo. * See note (2), page 46. tSee Theorem IIa. t See pages 43-46. ~ PP. 377-388. 83 84 UNIVERSITY OF MISSOURI STUDIES To apply Theorem II,* we write: Al Ai - A _S1 =; Si B- - B > I, B1 Bi - Bi-_ al(n) = ai(n) = > I. Bn B, Since ai(n) = I, i=l and since it follows from the hypotheses that L ai(n) = o, and ai(n) 2 o, n=-00 we may apply Theorem II,* and say: If Lsn =s or +,00 n=-00 then n A, A i"A A A L 2ai(n)si - + L E B = + L s or + oo. Wnoo i=l B 1 n=oo i=l -n n=o0 Bn THEOREM O: If the sequences (sn), (tn) converge to the limits s, t, then L sltn + s2tn1 + + * + ntl L = st. n=oo n Here choose sequence tn-i+l Sn = n, and ai(n) = nt Now I t Lai(n) = L - rn= In =00 n t and ai(n) =L I t- + t2 + - -+ tn L 2ai(n)-L - = n=oo i=1 n=oo t 'since Ltn = t. *Also Th m n=om * Also Theorem Ia. DEFINITION OF SUM OF A DIVERGENT SERIES 85 Furthermore, Itl I + t1 t2 |+ *** + tn | I nk k,=1 a ) t =n tn tsince [tnI < k, because Lt = t n=o00 Hence, applying Theorem I, we obtain LY^n n, \ tn-i+l I Sltn 2-+ S2tn-1 ' * ' + Snti L ai(n)si-= L s- i's =.- L n=oo i=1 n=00oo =l nt t n=00 n = s, so that s lt + S2tn-l + *- + Snti s L -- = s t. n=00 on We shall now prove Theorem L, which we stated on page 35 without proof. THEOREM L: If YCn is a divergent series of positive terms, then coso + C1S1 + C2s2 +.. + CnSn _ so + s5 + S2 +.. + Sn L n=oo.;^ n=o 0 n provided that the second limit exists and either (a) cn steadily decreases, (b) cn steadily increases, subject to the condition ncn < (co + cl + * * + cn)K, where K is a fixed number. I n either case, we put So + 5 +. + Sn n+ I (i + I)(ci - Ci+X ) ai(n) = -, i t n, =0o (n + I)Cn an(n) = S Ci i=O 86 UNIVERSITY OF MISSOURI STUDIES Since by hypothesis L ci = 00, n=00,=0 we obtain L ai(n) = o. n=oo Again L - a(,)= L (co-Cl) +2(Cl-C2) +.. +n(c-1-cn) +(n+I)C L E ai(n)- L --- ni=oo i= 0 =oo E Ci i=O Furthermore, in case (a), ai(n) > o, since by hypothesis cn+l < cn; and in case (b), n I Elai(n)- I [(cl - Co) + 2(C2 - C) + * n(c* - cCn-1) i=o0 V i= - + (n + I)Cn] since by hypothesis C,+l > cn; i. e., On I lai(n)l = [- (Co + Cl+ * * + Cn-1) + (2n + I)Cn] i= — O ~ i i=O 2 (n i)C, ncn(n+ ) — I +-2(n -- I)C < 2 < 4K. ES n Ci i=O i=O Hence in either case (a) or (b), we have: L ai(n)i = L n [(Co - Cl)ao + 2(Cl - C2)-1*... n=oo i=O n=oo i + n(c,_l - Cn)an-l + (n + I)cnon] I = L -n [(co - c)S + (C1 - C2)(so + S1) + '' n=oo i=O + Cn(SO + S1 + * ' * + Sn)l =L =..L L so+SI+ s+- +Sn = L - =L on =L n=oo \ i=O n=oo?=oo n \ g. DEFINITION OF SUM OF A DIVERGENT SERIES 87 This theorem is a special case of the following more general theorem: THEOREM P: If Zbni c,n are two divergent series of positive terms, then n n E cisi E bisi i=O t=O provided that the second limit exists and that either (a) c, /bn steadily decreases or (b) c,n/b steadily increases subject to the condition Cn bn n 0<K n E Ci Z, bi, i=O i=o where K is fixed. Here we put n E bii i=o O'n = n i=0 so that bns = (bo + b1 + * * + bn)an - (bo + b1 + + bn-l)n-l, and set Ci c+l \ bo + bi + * * * + bi a,(n) = bi bi+l JCo ' + Ci -+ ** + Cn cn bo + bl + ' + bn an ( nbn Co + c- + *- + Cn In the first place, since n L Eci = + 0, n= oo i=O it follows that L ai(n) = o. n= oo 88 UNIVERSITY OF MISSOURI STUDIES Also L a(l) = L I t(c _K +l)(bo + b, + * + bi) I. n= oo -=0 n=oo '6=0o bi bi+l, i=O Again in case (a), ai(n) > o; and in case (b), n I E la(n) = --- -(co+cl+- +cn-l) Zc i Cn c<,=o ] *=~q + n (2b~-+2b1+ * * * +2bnl-+bn) - I + 2 Now we have: bo + bi +. + b,, cn < 2K. I n (CO C1 C1 C2\ o -b bo * so+b -b J boso + bis X (bo+bl) b - b + ' bo + bi LE ai(n)oT = L n=oo i=O n=oo 9=, i=O n=L - o b ( iio ) b b o2 +(b-) boso+b0+... i=O ( C -1 C bn-l bn + b+ - b (boso+bls+~.* + b,_.s._-) + bn (boso+blsl+ -- *. +bnlsnl+bnsn) I -= L n [CoSo + CSi + * * * + Cn-ISn-1 + CnSnL] n=ox Z i -=0 Thus in either case (a) or (b) we have the theorem established, since n L E ai(n)o=i L o-n n=oo i=O n=oo whenever the latter exists. ~ IO. CONCLUSION In this concluding section we propose to recall some of our main results, to show wherein they fall short of being complete, and thus to formulate the problem which remains to be solved. Our results of ~ 3, concerning averageable sequences, are not of great value, since they involve a knowledge of the existence of certain limit points before the question of the existence of the averageable limit could have any significance. On the other hand, Theorem 3 is found useful in practice, in showing that certain classes of averageable sequences are summable (C1). Though we have discussed more general definitions, we shall confine most of our consideration in this section to the Adefinition of evaluability. It need hardly be pointed out that one of the inadequacies of the A-definition is that it may not be unique; that is, two specific sets of numbers ai, and bin, both satisfying the conditions of the A-definition, may give different values to the same sequence. Thus the sequence si = (- I)+1 log i has two different* values for the two different definitions: I I.-' ]+(-i)r+' I I ~ n *log =-..., = I. In fact* the former definition gives the sequence (si) the value o, while the latter gives it the value I. Two questions accordingly present themselves. First: given two A-definitions, what is a sufficient condition that one defi* See p. 38. 89 90 UNIVERSITY OF MISSOURI STUDIES nition be a generalization* of the other? Secondly: under what conditions are the two definitions equivalentt in scope? We shall now consider each of these questions in turn. The answer to the first question will be made clear by a few propositions. THEOREM 26: If In = alnsl+a2nS2+ * * +annsn Sn = 1 +22+ +nn = sln2l 22* * +nnbsl+b2nS2+'* * +bnnsn, where ain satisfy conditions of A-evaluability,$ n bin = I ain o, Lin =, iz= n-=00 and if LIn = s, then LSn = s. n=oo n=oo To prove this, we observe that by substituting the expression for;i in the first expression given for Sn, and equating the resulting coefficients of si to the coefficients of si in the second expression for Sn, we obtain ainann + ai n-llan-i n + ai n-2an-2 n + * * + aiiain = bin. Adding these equations for i = I, 2, — * n, we get: n n-1 ann E in + an-1 n E ai n-1 + * * i=l i=l i=.7~~~j~~ n + ain ai + * + aln *, = a bin i=l i=l or ann + ani1 n + * * * + a;in + + ain = I. Thus the numbers ain satisfy all the conditions of Theorem I; and our theorem is proved. * Thus, if A is (Ck) and A2 is (Cz), then A2 is a generalization of A1, if I > k; i. e., if when A1 gives to (Sn) a sum, then A2 will give to (Sn) the same sum. t Thus (Hr) and (Cr) 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 s. t See page 49, including footnote. DEFINITION OF SUM OF A DIVERGENT SERIES 9I Now assuming ann = o, and considering the formula ainoann + ai n-ilan-1 n + '* * aiji in = bin as n - i + I linear equations in the (n - i + I) letters ain, ai+l, n ' ann; the determinant of the system of equations is ann 0.. * * 0 an-1 n an-1 n-1... 0 = ann^n-1 n-1..* all 0, a1n al n-1 a11 so that ann 0 * * O bnn ain aiiai+1 i+i... ann. ai+l n ai+l n-1... ai+ i+ bi+1 n ain ai n-1... ai i+1 bin D aiiai i+ i+1 ~ ann We may then restate the previous theorem as follows: THEOREM 27: If ain, bin are numbers satisfying conditions for A-evaluability, and D aii =* o, ain = > 0,*:L aoin 0; aii.'' ann n=oo and if n L ainsi = s, n=oo i=l then L E binSi = s. n=oo i=I * See p. 49, footnote. UNIVERSITY OF MISSOURI STUDIES 92 In particular, let ain be the Ceshro coefficients for (Cr), r(r + I). (r +n -i - I) Cr+n-__-, n-i (n -i )! a Cr+n-l,n-1 (+ I)(r +2).. (r-+n- I)' (n- I)! so that on evaluating the determinant D, we obtain ain = a (bin -rbi+l, n + (r- I) b+2, + (-I)rbi+r, ) or, using the notation r(r- I) (bin - bi+, n)r = bin- rbi+, n + ( 2 )bi+2 n + (- I)rbi, n IX:~~~~~~I I 2 =in (bin-bi+l, n) r [(bin- b~+, n)r-1-(bi+l, n-bi+2, n) r-l]. aii aii It is evident that La in = o if Lbin = o; =oo n= oo00 hence we may say: THEOREM 28: If bin, corresponding to a definition B of evaluability, satisfies the condition (bin - bi+, n)r > o,* then if the sequence (sn) is summable (Cr), it is also evaluable according to the B-definition. If we let bin be the coefficients for summability (Hr), i. e., nbin - (i, n)r-2 (i + I, n)r-2 _ (n, n)r-2 nbi,, = (i, n)r_l = - - + + -- i +.+ I n where I I (i, )1 = i+ I 't + + ' then (i, n)1 - (i + i, n)1 = 7 (i, n)p-l (i, n)p - (i + I, n)p = (in) n * The condition I(bi,n - bi+1l n)rl < K is sufficient. i=1 DEFINITION OF SUM OF A DIVERGENT SERIES 93 Now n(bin - bi+-, n)l = [(i )r-i - (i + I, n)r-] = )r (, n),-2 (i + I, n)_-2 (i, n)r-2 + (i, n)r-3 n(bin- bi+l, n)2 = i - I ( i + tI-) Assume pl(i, n)_r- + P2(i, n),._3 + ~ * + p (i, n) r-j-i_ n(bin,- bi+l n)i = (, i > 0. Then n(bin - bi+l, n)i+l = n [(bin - b+l, n) - (bi+l, ~ - bi+2, n)i] ipl(i, n)r-2 + (P1 +jP2)(i, )r_-3 +. + Pj(i, n)r-j-2 i(i + I) (i + j) _ (i, n)r-2 + (2(i, n)?r-3 + + j+* * (ii, n)r-j-2 i(i + I).. (i +j) ai > 0. Hence by mathematical induction b, ) =pl(i, n)r-2 + P2(i, n)r-3 + * * + pi(i, n)r-j-1 n(bqn - bi+l,.)i = - - +) **(+ - ) pi > o, and accordingly (bin - bi+1, n) o. Thus, by our last theorem, we may say: THEOREM Q: If the sequence (sn) 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, I9I0, and by Schnee, Math. Annalen, Vol. 67, I909. The converse which has been first proved by Knopp, inaugural dissertation (Berlin, I907), can also be proved by using Theorem 29. t Quarterly Journal, Vol. 38, I907, 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 If ai > o, bi > o, n n A4n= ai, Bn=Zbi, LBn =oo, L An=o, i=1 i=1 n=oo n=oo and if either bi bi+l ai ai+l or b < b+ and < K K >o, ai ai+l Bn An and if also L alsl + '' - anSn n=oo a + * * * + a then blsl + * * * + bnsn n=oo bi + * + bn Let ai bi ain, - A bin =- B and an 0 * 0 bn an-1 an-1 ~. 0o bn_1 Oain =..... ann ''' aii ai+j ai+l *~ ai+1 bi+1 ai ai... ai bi I A b bi+l ] BnL i ai ai+lJ Since LBn =oo, it follows that L ai = o. n=oo If further bi bi+l ai a~+l DEFINITION OF SUM OF A DIVERGENT SERIES 95 then ain > o. If bi< bi+i ai ai+l then [(52 bi )Al_(b3 b)A2 z'=l '~l B, \a2 a\, a3 a2/, (bn bn-l\ A,. bn A an an-1 an = - - b[ - bn-1] + b. (A,_ + An),n Bn an bn An bn an = I +2 - <- I +2K, since - <K * Bn a Bn An Thus Hardy's theorem is proved* by applying Theorem 27.t 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 summability be a generalization of the other. Though these sufficient conditions are fairly simple, and prove useful in leading to important theorems, it would seem extremely desirable to have sufficient conditions that D > o.1 To answer the second question, we need only observe that if we can prove by Theorem 27 that definition (A) 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. 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