Abel's TJieurcDi and its converses 131 



are each necessary for the existence of 



lim s^^^ = / ; 



(tnd, taken together, they are suMcient. 



This theorem includes the analogue of 1 : 



Theorem 4. If lim s\^ =1 exists and is finite, then 



lim 2 aK*" = I- 



x^\ 1 



It is easy to verify that lim ij,^* = lim 6-J^* if \ = 1 or 2 : for 



higher values of X this relation certainly holds if both limits exist, 

 as follows from Theorems 1 and 4. 



In Part II, I propose to extend Theorem 3 to certain other 

 mean values; and Part III contains some general remarks about 

 the converse of Abel's theorem. 



Part I. 



]. In the researches which follow I have to make use of the 

 following theorems. 



n 



Theorem 5. // lim 2 a« = lim s,,. = I (finite), 

 and Uk is positive and 



n, 



lim 2 6k = lim t^ = ^ , 



1 " 

 then lim ~^b^s^= lim s„ = / (-1). 



This theorem is due to Stolz*. 



Theorem 6. Suppose that b^ is positive and 26« divergent; 

 a)id let D be the region defined by 



p<2cosi|^ {\ylr\^\lr^<^'Tr), 



where i — x = pe'''^. Further suppose that 



tb,\x''\/\^b^x''\<G, 



where (f is a finite constant, for all values of x inside the region D. 



* Bromwich, Injinitc series, p. 378. 



10—2 



