Abel's Theui'em cuid its converses 



143 



which scries of relations might be continued. They show (in con- 

 junction with Theorem 5) that lim s' = / whenever lim s — /, 

 from which we deduce 



Theorem 17. //" lim s|J' = I exists <incl is finite, then 



lim^ii,^ = 0. 

 Thus the second condition of Theorem 16 is necessary. 



12. We have also 



7 Pn—1 



'^n—i 



and thus 



V ,, ...K _ ^ ' h + ^ 'i" ,.K _ ^ ^« + l '/« " ^A-l „.K 



1 1 Ok+1 2 fx f^K 



II -\ 



In. 

 1 i«- 



= (1-^) 



V^!^±LZLil,,,K_i^ 



^K + l ^«.+ l 



^, #« - ^«_i ^« - (y«_i 



+ Z ^ — T-?— w'' 



■1 t^ b^ 



+ 



'Jn+i'^ ^n-^ 



^n+\ bn+i 



[ 1 K+2 I t>«+l &«+2 ^J 



+ 1 h ^ ^ i h * • 



"n+1 "n+i 2 f/f Ox 



Now the series 



has a radius of convergence at least as great as 1, since lim -=^ = 



f?t— 1 



and =■ -^ tends to a limit or oscillates finitely. Thus 



n-1 ba •' 



lira p X'' = 



