EXPOSITION OF THEORY OF POWER SERIES. 
227 
"With regard to </> (#), the condition of absolute convergence is — 
n -A-rj, 
X < lim. 
’ + 1 A, 
n + 1 
or 
i.e 
X < a, since lim. 
n + 1 
— a < # < a 
= 1 
as before. 
During the interval of absolute convergency we can write (from 
th. II.)— 
/•* 
<f> (x) dx — a x x + a 2 x 2 + . . . + a n x n + . . * 
equality, whose meaning is that (x), is the derivative of — 
u = a 0 + a x x + a 2 x 2 + . . . + a n x n + . . . 
therefore — 
</> (a?) = 
du 
dx 
This proposition is very important, for, as Weierstrass has pointed 
out in the remark given by Heine, it may occur that, although a series 
is convergent, the series formed by the derivatives may be divergent, 
and therefore represent nothing. Such is the case for the convergent- 
series — 
sm-g sin. (2* x) 
1 9 2 
+ 
sin. (3 3 x ) 
+ **•• + 
sin. (n 7 \x) 
n 3 
+ 
The series, deduced by derivating its terms, is divergent. 
In short, the series — 
f(x) = a Q + a, x + a, x 2 + + a n x n + 
absolutely convergent between — a and + a, defines during that 
interval a continuous function of a?, capable of derivation^ and 
integration — 
= + + na n x n ~ l + 
/ x 
v , , a x x 7 a.x* 
f (x) dx — a a x + — + + 
v,n + i 
a n x 
+ + 
n + 1 
and we have, for instance, from the continuity — 
lim./ (#) = f (lim. x) 
during the interval of absolute convergency. 
