Abel's Theorem and its converses 145 



15. The proof of the converse begins with equation (20), which 

 is valid since lini ~ = 0. Therefore 



X^-l 1 f/C + 1 



This is equivalent to the first condition of Theorem 15. But the 

 second is satisfied too, viz. 



lini yit. ^A^i^i ^ n„, 1 [-^^^^ -())+...+ (fy, - r/„_,)] 



«-*-x hi 2 tie M-*-c» f)i 



= lim '^ = 0. 

 Thus lim S '^'^^'~'^'' = ^. 



and, by equation (21), 



lim s[y = I, 



which completes the demonstration. 



The conditions (14) and (19) imposed on the numbers b^ are 

 not necessary but only sufficient. The conditions necessary and 

 sufficient would depend also on the coefficients «« of the power 

 series considered, so that for a given series 'S^a^x" a given set 6^ 

 )nay be admitted which must be excluded for other series ^CkX". 



Part III. 



16. Theorem 2 is in a sense a perfect converse of Abel's 

 theorem, from which all these researches originated. 



Series for which Abel's limit exists may be divided into two 

 classes, those which are convergent and those which are divergent, 

 series for which the limit does not exist being excluded. 

 Theorem 2 shows that the first class consists of those, and those 

 only, which satisfy the condition 



lim -i«ct, = (22). 



The second class consists of those, and those only, which do not 

 satisfy the condition. 



The condition (22) is satisfied, in particular, if 



lim nau = (23) 



VOL. XIX. PAliT IV. 11 



