generally uniform; that is, it depends on
x
. In this regard, recall
also a useful result given in Exercise 55.7.25
57.2. Radius of Convergence.
The set
S
on which a power series is
convergent is an important characteristic and its properties have to be
studied.
Lemma
8.3 (Properties of a Power Series)
.
(i)
. If a power series
∑
c
n
x
n
converges when
x
=
b
= 0
, then it converges whenever

x

<

b

.
(ii)
. If a power series
∑
c
n
x
n
diverges when
x
=
d
= 0
, then it diverges
whenever

x

>

d

.
Proof.
If
∑
c
n
b
n
converges, then, by the necessary condition for
convergence,
c
n
b
n
→
0 as
n
→ ∞
. This means, in particular, that, for
ε
= 1, there exists an integer
N
such that

c
n
b
n

< ε
= 1 for all
n > N
.
Thus, for
n > N
,

c
n
x
n

=
c
n
b
n
x
n
b
n
=

c
n
b
n

x
b
n
<
x
b
n
.
which shows that the series
∑
c
n
x
n
converges by comparison with the
geometric series
∑
q
n
, where
q
=
x/b
and

x/b

<
1 or

x

<

b

.
136
8. SEQUENCES AND SERIES
Suppose that
∑
c
n
d
n
diverges. If
x
is any number such that

x

>

d

,
then
∑
c
n
x
n
cannot converge because, by part (i) of the lemma, the
convergence of
∑
c
n
x
n
implies the convergence of
∑
c
n
d
n
. Therefore,
∑
c
n
x
n
diverges.
This lemma allows us to establish the following description of the
set
S
.
Theorem
8.27 (Convergence Properties of a Power Series)
.
For a
power series
∑
c
n
x
n
, there are only three possibilities:
(i)
The series converges only when
x
= 0
.
(ii)
The series converges for all
x
.
(iii)
There is a positive number
R
such that the series converges if

x

< R
and diverges if

x

> R
.
Proof.
Suppose that neither case 1 nor case 2 is true. Then there
are numbers
b
= 0 and
d
= 0 such that the power series converges for
x
=
b
and diverges for
x
=
d
. By Lemma 8.3, the set of convergence
S
lies in the interval

x
 ≤ 
d

for all
x
∈
S
. This shows that

d

is an
upper bound for the set
S
. By the completeness axiom,
S
has a least
upper bound
R
= sup
S
. If

x

> R
, then
x
∈
S
, and
∑
c
n
x
n
diverges.
If

x

< R
, then

x

is not an upper bound for
S
, and there exists a
number
b
∈
S
such that
b >

x

.
Since
b
∈
S
,
∑
c
n
x
n
converges by
Lemma 8.3.
Theorem 8.27 shows that a power series converges in a
single
open
interval (
−
R, R
) and diverges outside this interval. The set
S
may or
may not include the points
x
=
±
R
. This question requires a special
investigation just like in Example 8.38. So the number
R
is character
istic for convergence properties of a power series.
Definition
8.11 (Radius of Convergence)
.
The
radius of conver
gence
of a power series
∑
c
n
x
n
is a positive number
R >
0
such that the
series converges in the open interval
(
−
R, R
)
and diverges outside it.
A power series is said to have a
zero radius of convergence
,
R
= 0
, if it
converges only when
x
= 0
. A power series is said to have an
infinite
radius of convergence
,
R
=
∞
, if it converges for all values of
x
.
The ratio or root test can be used to determine the radius of con
vergence.
57. POWER SERIES
137
Corollary
8.3 (Radius of Convergence of a Power Series)
.
Given
a power series
∑
c
n
x
n
,
if
lim
n
→∞

c
n
+1


c
n

=
α
=
⇒
R
=
1
α
,
if
lim
n
→∞
n

c
n

=
α
=
⇒
R
=
1
α
,
where
R
= 0
if
α
=
∞
and
R
=
∞
if
α
= 0
.