A power series 
will converge only for certain values of 
. For instance, 
converges for 
. In general, there is always an interval 
in which a power series
converges, and the number 
is called the radius of convergence (while the interval itself
is called the interval of convergence). The quantity 
is called the radius of convergence because, in the case of
a power series with complex coefficients, the values of 
with 
form an open disk with
radius 
.
A power series always converges absolutely within its radius of convergence. This can be seen by fixing 
and supposing that there exists
a subsequence 
such that 
is unbounded.
Then the power series 
does not converge
(in fact, the terms are unbounded) because it fails the limit
test. Therefore, for 
with 
, the power series does not converge, where
 |
(1)
|
 |
(2)
|
and 
denotes the supremum limit.
Conversely, suppose that 
. Then for any radius 
with 
, the terms 
satisfy
 |
(3)
|
for 
large enough (depending on 
). It is sufficient to fix a value for 
in between 
and 
. Because 
, the power series is dominated by a convergent geometric
series. Hence, the power series converges absolutely
by the limit comparison test.
