22J 

 also limited, and therefore: 



n 

 Heiioe by (16) we conclude: 



lim.[d'\o^'')-ö;i'+'^]=0 (19) 



Now it is a well-known tlieoreni that /hn. H„ (t) = s implies 



nrroo 



liin. Hn (/) = *•;') hence we deduce from (19) with the aid of (9): 

 hm.[H,i {a ) — Hn {o ^ )] — 



from whicii we conclude: 



If lim. hIP (</*■■') = 0, we have also Vim. öL'+'^ [o^''~^^) = 0. (20) 



n=zoo n=X' 



By hypothesis we have: /i)n. n . A[,r — ^J,''li] = Ü or by (J3) 

 Hence by (19) 



lim. On =0. 



fi) (p-i) 

 lim, Hn (o ) =^^ 



n:=cc 



and applying (20) we get successively : 



tirn. Hn [O I ^ W 



lim. Hn lo J = y 



lim. hIP [o'' '^] = . (21) 



Höi.DKH has proved') that if lim. H„{t)^k, then we have also: 



lim. 2 {tn-\-i — tn) ■>'" = h. 



2-_>l 



In virtue of this theorem we have l)y (21) 



(P-i) iP-i -. „ 



or by (18) 





lim. (1 — x) . w' (x) = 

 thus by (J 7): 



') See Bromwich, Theory of infinite Series, p. 383. 

 *) Matliematische Annalen, Bd. 20 (1882), p. 535. 



15* 



