144 Dr Kieiiad, E.dennwns of 



for every x < 1 , arul therefore 



i a.,- = {i-xfi p" ^« + (1 - ^) V ^±LZL^J ^ ^ 



Taking account of the conditions (19), it follows from Theorem 6 

 that 



lim(l -xf%p'a:- = 0, 



x^-\ 1 Ok +2 



and Iini (1 - x)t ^tir_^«+> f+1 ,.. = 0, 



X^\ 1 Ok-I-1 0^+2 



SO that lim % a^x" = lim S ^ "^^ ~ ^^ a;'^^^ (20). 



13. Lastly we have the identity 



14 If lim 6-|/^ = ^ exists and is finite, then, by (21), i ^"^' ""^^^ 



1 ^K + l 



converges to the sum I. Therefore by Abel's theorem 



and since (Theorem 17) lim ^ = 0, equation (20) is valid, and 

 thus 



lim 1a^x'' = l. 



x^l 1 



We have therefore 



Theorem 18. Let the coefficients h^ be chosen so as to satisfy 

 the conditions (19). Then, if lim 6'J/* = I exists and is finite. 



lira 2 a^x" = I. 



x^\ 1 



The hrst condition of Theorem 16 is consequently necessary. 



