PROCEEDINGS 



OF THE 



Extensioits of Abel's 'Theorem and its converses. By Dr A. 

 KiENAST, Kiisnacht, Zurich, Switzerland. (Communicated by 

 Mr G. H. Hardy.) 



[Received 26 September 1917.] 

 Introduction. 

 Abel pruved in 1826 the theorem : 



n 



''If liui !£ (^ ejists and is finite, then 



lim % a^x" = lim ^ a„." 



Let us write 



Then H(3lder* proved in 1882 



Theorem 1. If lim t^^ exists and is finite, then 



.(1). 



lim S ai^x" = lim t]^ . 



* Bromwifh, Iiijinite scries, p. 313. 

 VOL. XIX. PART IV. 



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