Abel's Theorem and its converses 147 



««K > — A" of Theorem 19 we have \X t^a,, < K, a condition which 



I 1 

 allows Kti^ to tend to infinity in either direction. 



That such cases exist, in which S «« is convergent, is shown by 

 the fact that 



t-— (0<e<</)<27r-e) 



is convergent if t^ is any function of k which tends steadily to 

 infinity with k. 



18. A similar result can be obtained from another theorem of 

 Messrs Hardy and Littlewood, viz. : 



Theorem 20. If f{x) = S a^af is a potver series with positive 

 coefficients, luul f{x)^ ^ as x^\, then 



n 



2 a« ~ /I. * 

 1 



From this theorem it is possible to deduce Theorem 19 (see 

 above) of the same authors. 



Now the hypothesis is equivalent to 



CO 



lim (1 — x) f{x) = lim % (a^ — a^-i) x" = 1, 

 and the conclusion is 



lim -^a^= hm - S J 2 (a^ - «a-i) h = 1. 



M-*M '>l 1 M-*.0O ^ 1 ( 1 j 



Thus Theorem 20 is equivalent to 



CO n 



Theorem 21. If limXb^x" = 1, and if the sums «„ = ^ 6« 



x-*l 1 1 



are alt positive, then 



1 'i 

 lim - 2 6'k = 1. 



»-*-oo 11 1 



Here again is a condition which, in case the series converges, 

 does not prevent the real numbers Kb^ from tending to infinity in 

 both directions. 



* G. H. Hardy and J. E. Littlewood, I.e. See also E. Landau, I.e., § 9. 



11 — 2 



