90 
SERIES 
2. Thus we conclude that the set of points for which P con 
verges form an interval (a — p, a + p) about the point a, called 
the interval of convergence ; p is called its norm. We say P is 
developed about the point a. When a = 0, the series 1) takes on 
the simpler form ^ + v + ¥?+ .,. 
which for many purposes is just as general as 1). We shall 
therefore employ it to simplify our equations. 
We note that the geometric series is a simple case of a power 
series. 
86. Cauchy's Theorem on the Interval of Convergence. 
The norm p of the interval of convergence of the power series, 
P = a Q + ape + ap? + ••• 
is given by 
- = lim V a„ 
P 
«„ = \a* 
We show II diverges if £>p. For let 
p I 
Then by I, 338, 1, there exist an infinity of indices q, ¿ 2 
which 
V^ n >/3. 
Hence 
and thus 
since >1. Hence 
«t„ > /3 ln , 
for 
n 
is divergent and therefore n. 
We show now that n converges if |<p. For let 
«<!<'■ 
Then there exist only a finite number of indices for which 
VcT n >/3. 
Let m be the greatest of these indices. Then