Abels Theorem and its converses 141 



NoAv Theorem 2 can be applied to the series S --^ — - j-^ m^", 



provided that 



hiu -1,K — , — J = (lo). 



Assuming for a moment that this condition is satisfied, 

 Theorem 2 leads to 



lim ^i<±}-zi<l^ = i. 



Il^-X I ix *K+l 



and (16) gives finally 



lim Sn = I, 



proving the theorem, which is the analogue to Theorem 2. 



Condition (18) depends on the 6's as well as on the as; but 

 since lim J-^ = 0. it will certainly be fulfilled when 



-Zk -— 



n I Ik 



tends to a limit or oscillates finitely. For, e being given, we can 

 choose K so that 



I ^ ^A+, -t^ Pk ^1 "^^ ^ tK+i -tK pK , e V ^ ^A+i - tK 



— ^ A. < - Zi /^ 7 i — A, - . 



n 1 Ik t\+i n \ Ik t^+i " < t^\ 



We may suppose, for example, that 



tn = ri''; \ogn; log log 7i;... 



10. Adding to the notations used hitherto 



1^7 (1) 



/ .1 A A-l 71 ' 



^lbj^, = sf, 



'"n 3 



S Ox ^ = CJn , 



2 ^A-l 



and restricting the choice of the numbers b^ not only as done in 8, 

 but further by supposing that the two limits 



lin, L ^1 , lim h+ip^h+^ ^1 (19) 



