458 



(^,^(..)=l[^'„"-4-i]..r". (1) 



GO 



V. (•'') = -^ "n •*" (1«) 



1 



(p^{x) + {l-:c).,f^{a;) = (pj^_^(x) .... (2) 

 n.[A„ — A„^\\=Au — A„—i . . . . (d) 



00 



With the aid of (2) we have proved ^) that Hm . i:a„w" = s implies 



Now, if moreover: 



Urn. (f (x) ■=. 4. 



r ^W ^W 1 I ^ 



n . [yl;, — 4„_i] 1 < c, 



we have by tlieoreni 1 tiiat i[^(*) — -^^l^il <^oiiverges to .«, i. e. 



lim. An ^ « 



or: -S'd,, is summable of order k. 

 Since 



.lt)_^l + ^-12 + ■ . ■ - + ^» 



yl„ — 



11 



we infer from | JiA'— 1)| <^ c : | J'.*'| <^ c and bj (3): 



n lA^^ - A^\\ 1 < 2c. 



Hence we see tliat 



\Ai I <C « and /«m. ^ a„ .«" :^ S 



J— ».l 1 



imply that ^a» is summable of order /;. 



We use in \ 2 the particnlar case that ^- = 1. It tlien runs: 



00 



Theorem 3: If lim. ^ anX"=:s, and \sn\ <C C '''"''i lim. r>n = s, 



ivhere 



^'i + -^i + • • • + S" 



On = . 



n 



1) Loc. cit. p. 222. 



I 



