Abel's Theorem and its converses 133 



Thus the series S r*^' , ,.r'^"^^~" converges if Lt I < 1 ; the same 

 is the case with ^^ 



or i r(\+;\«,-'^^^-l 



J) + A — 1 



Differentiating the last series (X — 2) times, we obtain 



SO^ + 1 )0; + 2) ... (^9 + X - 2)7-<^+;':;,a''^ ; 

 which gives 



00 



Theorem 8. // the radius of convergence of P (a-) = 2 a^.r" is 



1 

 unity, then for every \x\<\ 



r (A + k) n+A-2 r. 



hm ?■ , , , ./• = 0, 



/l+A — 1 ' 



hm - r ic = 0, 



lim (>i + 1 ) (7? + 2) . . . (n + \ - 2) r^+^^l^ «^" = 0. 



2. The demonstration of Theorem 8 depends on certain 

 identities. The formula 



' n " 



leads, by successive summation, to the series of equations 



(6). 



(2) (1) 1 (2) 



n I) j^ /( 



^(A) _ _(A-1I _ J- ^.(A) 



If lim s =1 exists, then, bv Theorem 5, lira s . also exists 



and is equal to /, and therefore one of these identities gives 



r 1 (A+i) r\ 

 Inn - ?' = 0. 



Theorem 9. If lim .s^^' = I exists and is finite, then 



n-*-x 



r 1 (A + l( n. 



lim 7" = 0. 



7l-»-X " 



Thus the second condition of Theorem 3 is necessaiy. 



