EXPOSITION OF THEORY OF POWER SERIES. 
225 
THEOREM III. 
If u be convergent for x = x of it will be absolutely convergent for 
all values of x smaller in absolute value than x 0 . (See Abel toe. cit ., 
th. II., CEuvres, p. 222.) 
Let A represent the absolute value of a 
and X „ ,, ,, x. 
If the series is convergent for x = x Q its terms must be ultimately 
smaller than anv given quantity a, however small— that is to say, we 
must have A M X D ” < a, whatever be n ; therefore the series — 
where — 
(i) 
A 0 + A t X + i . . . -b A a X* + . . . . 
x<x Q 
is convergent as having its terms respectively smaller than those of 
the obviously convergent series — 
a + a 
X 
x n 
+ a 
+ .... 4- a 
+ . . . 
But the convergence of (1) means, by definition, the absolute 
convergence of u , so that our proposition is established. 
THEOREM IV. 
If u be absolutely convergent for x = x OJ it will be uniformly 
convergent for — 
— x Q < x < x Q 
Urn its included. 
From a well-known theorem on convergence of series whose terms 
are positive integers,* to say that u is absolutely convergent for 
x = x 0 means that we can take n sufficiently great to have 
r„(x 0 )=a„ + ix;‘ +i +a„ + 3 x” 
smaller than e, e being a given quantity as small as we please, whatever 
be the value of^>. 
But taking X S X 0 we have — 
A-n + 1 
A-n + 2 
X n + 1 
o 
X n + 2 
o 
n + 1 
n + 2 
A ~V‘ n + 1 
+ 1 -A-o 
A + 2 
“w + 2 ^-o 
+ p 
X 
n + p 
(IJ 
A 
n + p 
•yrn + p 
P 
See foot-note, p. 218. 
