284 CHAPTER 5 Series and Differential Equations
5.3.1 Radius and interval of convergence
Our primary tool in determining the convergence properies of a power
series
∑∞
n=0
anxnwill be theRatio Test. Recall that the series
∑∞
n=0
|anxn|
will converge if
1 > nlim→∞
|an+1xn+1|
|anxn|
= |x|nlim→∞
∣∣
∣∣
∣
an+1
an
∣∣
∣∣
∣,
which means that
∑∞
n=0
anxnis absolutely convergent for all x satisfying |x|<nlim→∞
∣∣
∣∣
∣
an
an+1
∣∣
∣∣
∣.
The quantityR= limn→∞
∣∣
∣∣
∣
an
an+1
∣∣
∣∣
∣is sometimes called theradius of con-
vergenceof the power series
∑∞
n=0
anxn. Again, as long as−R < x < R,
we are guaranteed that
∑∞
n=0
anxn is absolutely convergent and hence
convergent.
A few simple examples should be instructive.
Example 1. The power series
∑∞
n=0
(−1)nxn
2 n+ 1
has radius of convergence
R = limn→∞
∣∣
∣∣
∣
an
an+1
∣∣
∣∣
∣ = limn→∞
( n
2 n+1
)
(n+1
2 n+3
) = limn→∞ n(2n+ 3)
(n+ 1)(2n+ 1)
= 1.
This means that the above power series has radius of convergence 1 and
so the series is absolutely convergent for− 1 < x <1.
Example 2. The power series
∑∞
n=0
nxn
2 n
has radius of convergence