Abel's Theorem and its converses 



1:17 



Hence 



Theorem 12. //' Xok-"''" Iki-'^ <inity as f(((Iiiis of convergence, 

 and if 



1 





then 



.(p) 



'^^ 1 



r " = lini 2, 



,(p+i) 



I 



Another identity is acquired by developing s\f (;? = p + l^ 

 /) + 2, ...) in the form 



P + 1 



P+2 



>)_JP) 1 



j(P-l) 



pVi 



p+2 



1 ^(p-1) 



1 V.*"-^' 



V/i(?W — 1) 



I 



"-1 1 ( \ 



VI (m + 1) 



(p) 



til en 



Theorem 13. If .s"' and r ''' are defined as in (2) and (3) 



S^'^= S 



.(P) 



If lim ,9''^* = I exists and is finite, then, by Theorem 5, 



II ^x 



r ('^+1) 7. 

 hm s = I : 



II 



ll->-orj 



and by Theorem 13 



S 1 



.(A+i) 



= /. 



.(^+1) 



Therefore by Abel's theorem 



TV -'- (A + 1) III V ■'- 



inn 2. - "Tvr ,r — z — 7 Ts 



x^i A+i ?H (?/i + 1 ) '" A+i m (m + \) '" 



On the same assumption, Theorem 9 gives lim -■^•^^"^ ■ =0; 

 and therefore Theorem 1 1 gives 



= /. 



lim 2 n.^x" — lim 2 



^,(A+l) « 



1X+1«(" + 1) " 



" 7 



