222 



ii)n.\.v.(r Ax) — a> (.n)] ^ 0. 



, '^ p—l ' ^p-i — 1^ ' 



Hence we infer lliat </'(p^i)(.«) and 'fp-;-i{x) oscillale about the 

 same region as *•—*]. Repeating llie argument for i = 1,2, . . p, 

 we see that 



X CD 



<r, (.(.•) = 2 a,, X" and 'f' (.^O ^ ^ [aT — Anil J .t" 



oscillate about tiie same region as .r —*■ 1. 



By iijpothesi.s we have //?». ?i [,4,i"'' — .4i/i-i] = 0; witli the aid ot 



tiieorem 1 we deduce that rppl-v) = ^ [An — A„-i] x" osciihvtes as 



1 



.c-^1 about the same region as A,i', =:^[^4„' — ^n--i | if ??( — ^ oo. 



Combining these results we see tiiat 2a„.);" oscillates as c — »• 1 



1 



about the same region as A,„^i') as Hi — *■ oo . 



I'roof of theoroii 3. 



1 



Lemma : If Urn. 'I,^_^ (■') = « and 'ik (•'-•) + (1— «) '/),. (^) = -V^. ^ (•^) 



then lim. <f {x) = s. 



Proof of thu lemma: If we solve the dilfei'eiitial equation we 

 become : 



./ (.c) = (1 -s) -* </x 4 C (\-x) 







Since h'ln. (fk—i{-v):^.i, it is possible whatever be 6]>0 to calculate 



x—^l 



a number è,<Cl ^^o that §,<^.r<^l implies: 



(l-.i) . — ^^ dx — a-x) — !^^— dx -I- (l-.t,-) — ^^ dx 



f, 



— (l_.r)^l-i d.f-f (1— .r) (/a;4-(l-.c) — ^^ rfx 



= / + i/ + /// 



L/?«. 7=0', therefore we can calculate a number S, ^ i, so 



