SECTION 5.3 Concept of Power Series 285
R = limn→∞∣∣
∣∣
∣an
an+1∣∣
∣∣
∣ = limn→∞(n
2 n)
(n+1
2 n+1) = limn→∞^2 n
n+ 1= 2,
so in this case the radius of convergence is 2, which guarantees that the
power series converges for allxsatisfying− 2 < x <2.
Example 3. Consider the power series
∑∞
n=0(−1)nxn
n!.In this case theradius of convergence is similarly computed:
R = limn→∞∣∣
∣∣
∣an
an+1∣∣
∣∣
∣ = limn→∞( 1
n!)
Å
1
(n+1)!ã = limn→∞n+ 1 = ∞.This infinite radius of convergence means that the power series
∑∞
n=0(−1)nxn
n!
actually converges for all real numbersx.
Example 4.We consider here the series
∑∞
n=0(x+ 2)n
n 2 n, which has radiusof convergence
R = limn→∞
(n+ 1)2n+1
n 2 n= 2.
This means that the series will converge where|x+ 2|<2, i.e., where
− 4 < x <0.
Example 5. Here we consider the power series
∑∞
n=0nx^2 n
2 n. The radius
of convergence is
R = limn→∞n
2 n·
2 n+1
2(n+ 1)= 2.
But this is a power series in x^2 and so will converge ifx^2 < 2. This
gives convergence on the interval−
√
2 < x <