Abel's Theorem and its converses 



180 



then 



r'''=l. 



1 n(n+l) 



This proof is actually given by Mr Tanber, and is therefore 

 the basis of the theorems of this paper. 



Part II. 



8. Let 6k denote the terms of an infinite sequence of positive 

 real numbers, which have the properties 



(1) lim S b^ = lim tn = oo 



.(14), 



(2) 



n 1 



tends to a limit or oscillates between finite limits. Then 

 Theorem 16. The two conditions 



QO 



lim S «'««"' = / (finite), 



x^l 1 



1 » 

 hm --l^AaA = 0, 



n-*-cc I'll 1 



are each necessary for the convergence of S a«, i.e. for the existence of 



II 

 lim %aK= I : 



and, taken together, they are sufficient. 



Abel's theorem states that the first of these conditions is 

 necessary. 



If lim Sn = I, then lim a„ = 0, and by Theorem -5 



1 " 



lim - S 6a.?a-i = lim 



ii^X ^'11 



1 n 1 n 



--2&ASA--S6AfO 



'« 1 f»i. 1 



/. 



The identity 



1 " 1 " 



f ft 2 f« 1 



n(^w gives, as a consequence of lim ,<?„ = /, 



1 

 Therefore the second condition is necessary too. 



nm ^ uxu\ 



