Professor Hardy. A theorem concerning summable series 305 

 The condition (3) is therefore equivalent to 



„+i(v+i)(^+2) ^irj ^''^- 



8. Suppose first that 2a„ is summable, to sum zero, i.e. that 

 '=0(71). Then 



s „ s ,,_j 



=^ lim Z 



,+l{v+l)(v + 2) rn^^ ,,+,{v + 1) {v + 2) 



Sf' o' /" - i C. 



m-1 



= lim ; ~ -? ^. - . ^^^— ^ +22 



= 2S 



_^ V(m + ])(m + 2) (/i + 2)(n + 3) ,,+i(v + l){v+2){v+^) 



S n 



„+i (z/ + 1) (v + 2){v + 3) (n + 2) (w + 3) 



-i»a=)-n^»C) :-;-^«)- 



Hence (5), and therefore (3), is a necessary condition for 

 summability. 



4. Suppose now that (5) is satisfied. Then, by (6), 



2 ^ , ^ii_ -^ =0 - ...(7). 



,+i(z.+ l)(j. + 2)(z. + 3) (7i+2)(n + 3) \n) ^ ' 



Writing 



<P^ = in + 2) {n + S)l^ (, + i)(,;2)(, + 3) ' 



we obtain 2^„ — s„' = (/i) (8). 



But 



"* s ' s ' 



4>, - (/>,_,= 2 {n + 2) ^.^^^^(^ + i)(,;2)(. + 3) ~ Vh 



_ -'<Pw ~ ^w _ /-I \ 

 ~ 71 + 3 ~ ^ ^' 



by (8) ; and therefore (/>„ = (?i) and ,s„' = o{n), so that the series 

 is' summable to sum 0. Thus the theorem is proved. 



5. In order to show that the theorem is not without applica- 

 tion, I apply it to the deduction of two known convergence criteria*. 



* See (for A) L. Fejev, 'La convergence sur son cercle cle convergence d'une 

 serie de puissances effectuant une representation conforme du cere le sur le plan 

 simple.' ComjJtes ReudKs, 6 .Jan. 1913, and 'tjber die Konvergenz dcr Potenzrcibe 

 an der Konvergenzgrenze in Fallen der konformen Abbildung auf die schliclite 

 Ebene,' H. A. Schwarz Festschrift, 1914, pp. 42-53; G. H. Hardy and J. E. Little- 

 wood, 'Some theorems concerning Dirichlet's series,' Messenger of Mathematics, 

 vol. 43, 1914, pp. 134-347: and (for B) G. H. Hardy 'Theorems relating to the 

 summability and convergence of slowly oscillating series,' Proc. London Math. Soc, 

 ser. 2, vol. 8, 1910, pp. 301-320; E. Landau ' Uber die Bedeutung einiger neuerer 

 Grenzwertsatze von Herrn Hardy und Axer,' Frace Matematijczno-fizijczne, vol. 21, 

 1910, pp. 97-177; M. CipoUa, ' Sul criterio di convergenza di Hardy,' Rend, dell' 

 Ace. di Napoli, ser. 3, vol. 26, 1920, pp. 96-107, 151-160. 



