130 



Dr Kienast, Extensions of 



In 1897 Mr Tauber, and in 1900 Mr Pringshciui, published the 

 following converse of Abel's theorem : 



Theorem 2. The two conditions 



lim 2 ««.'■" = I ( finite), 



1 " 



hm - 2 KCi^ = 



n-*'Oo ^^ 1 



are each necessary for the convergence ofX a^, i.e. for the existence of 



n 



lim ^a^ — l; 



n-*-ao 1 



and, taken together, they are sufficient'^. 



In the present paper I replace the means (1) by 



^ (2.) 



r;'+^^ = V - 7-(^) Ol = X + 1, X + 2, . . . ) 



I prove, in Part I, 



Theorem 3. Tlie tivo conditions 



•(3), 



lim S a^x" = I (finite), 



x-*l 1 

 Broniwjch, IntinlU series, p. 251. 



