146 Dr Kienast, Extensions of 



But this condition, unlike (22), is not a necessary condition for 

 convergence. 



Recent investigators have generalised the condition (23) in a 

 different manner. Thus Mr J. E. Littlewood proved* the theorem : 



" 2 a^ IS convergent, provided lim 2 a^x^" = A and 1 na„ \ < K." 



1 x^l 1 



And still more recently Mr G. H. Hardy and Mr J. E. Littlewood f 

 proved 



Theorem 19. If lim 2 a,,x'^ = A , and a„ >--K, then 2 a„ 

 x^i n 



converges to the sum A. 



But however interesting in themselves these two theorems and 

 their proofs may be they are less perfect than Theorem 2. For 

 the conditions j na,, j < K and na,, > - ^ are neither necessary for 

 convergence nor is either, together with \imta,x'' = A, necessary, 



nor do they characterise the non-converging series for which Abel's 

 limit exists. Their interest is in fact of a quite different character 

 from that of Theorem 2. 



It is not difficult to state similar theorems which are open to 

 the same objection but which give information in cases where the 

 last two theorems fail. 



17. The terms a^ of any sequence can be written in the form 

 «« = ^ » where t^ is subject to the same conditions as in Theorem 15. 

 This theorem then shows that 



CO ^ 00 ■ T 



" ^ -^ is convergent, provided lim 2 ^- a;" = ^ and lim -2c = " 



1 '''' x^l 1 ^K ,i-*oo tn 1 " 



Now the second condition is certainly satisfied if lim 2 c« tends 



to a limit or oscillates finitely. The only limitation'thus imposed 

 upon the order of magnitude of a« is that \c^\<K, i.e. that the 



order of /c | a« | does not exceed that of ^ . Instead of the condition 



T */' ,^- ^ittlewood, 'The converse of Abel's Theorem on power-series', Proc 

 London Math. Sac, ser. 2, vol. 9 (1911), p. 438. . -^ '<^c. 



_t G.H^ Hardy and J. E. Littlewood, ' Tauberian theorems concerning, power- 

 series and Dirichlet s series whose coeflicients are ijositive ', Proc. London Math '^nr 

 ser. 2, vol. 13 (1914) p. 188. See also E. LanJau, I)^.^./..,;"^ ^^'^^^^ 

 ciniger neuerer ErgebniHse der Funktionentheorie. (Berlin, 1916) pn 45 etsea ■ ihl 

 actual theorem is stated in § 9 and finally proved in § 10 (Die HardyiLittlewoodsche 

 Umkehrung des Abelschen Stetigkeitssatzes). 



