226 
PROCEEDINGS OF SECTION A. 
X 
since ^ < 1 ; wherefore by addition — 
■Xo 
a„ + 1 x; +1 ( x )’ +l +a „ +2 X ‘ +2 (|-)’ +2 + --- 
(I) 
X W + l’j 
0 
XV + ”-A T* + 1 4_ A V* + 2 
— + 1 -^-o d“ + 2 -Xo 
+ + A n + p X 0 
n + p 
or E re (X) S B- n (X.) whatever be the value of p ; 
therefore a 0 + a t x + a i x 2 + • . . . + a n x n + . . . . 
where XS X Q i.e., x < x Q in absolute value, 
or — x Q < x < x Q 
(the signs of inequality including the sign of equality) is uniformly 
convergent. 
In the case where the series is convergent (only) for x — x Q , we 
have seen (th. III.) that it is absolutely convergent, for 
where $ is a quantity as near x 0 as we please, but never equal to it ; 
so that (according to th. IV.) our series converges uniformly, for 
— £ < x < f . 
This theorem is due to Abel (loc. cit th. IV. and V., CEuvres 
p. 223), who has shown that the series is uniformly convergent even 
for x = x 0 . But his demonstration necessitates the establishment 
of a preliminary lemma (CEuvres, loc. cit., th. III., p. 222). The 
preceding demonstration is less complete, but easier to follow. 
THEOREM V. 
The series — 
< j> (x) = a x + 2a 2 x + . . . . + u a n x n + . . . . 
whose terms are respectively the derivatives of the terms of the 
series u , represents the derivative of this series u in^Jany interval 
whenever u is absolutely convergent. 
Eor u is absolutely convergent for 
A n + 1 X ,i + 1 
A* X" 
< 1 at the limit, or 
X < Lim. 
A” 
Let Lim. 
^-n + 1 
+ 1 
a ; then we have 
— a < x < a 
for the condition of absolute convergency of u. 
