^^^ Dr Kienast, Extensions of 



9. To prove the converse, we require two identities. If 



J^ hi 



n 



we have 



X a.af =f ^ + 2 - |^« - p,_,] x^ 



/ — 



1 ''K + l 



'«+r 



^« 



a' 







w-1 



1 ^K+l 1 t^ ^K + i 



Putting a; = 1, this gives the identity 



.(15). 



_\^t<±i-t^ Pk . p 



L 



t ^f 



''K+i hi 



•(16). 



If we suppose lim ^ = 0, it follows that 



lim^.r'»=0 



for every I .t | < 1 ; and, by Theorem 6, 



lim(l-^)i^,,- = 0. 



Now passing in (15) to the limit (first n -^ oo and then 

 a?— » 1 we find that if lim -^'=0, then 



lim 2 o„ a;" = lim 2 ""^^ -^JL.^" n^j) 



Theorem 15 starts from the assumptions 

 lim ^ = 0, 



n-^-x I'll 



CO 



lim S Ui^x" = I. 



x^\ 1 



The first assumption shows that (17) is available ; and this equation 

 gives, with the second assumption, 



hm z ~ -i-^ a* = /. 



x-^\ 1 t. L ,, 



