522
8.6. POWER SERIES
-2
-1
1
2
-10
-5
5
10
x
y
Figure 8.7: Graphs of partial sums of the power series
∞
k=1
x k
k 2
a particular curve on the interval (−1, 1), while growing without bound outside of that
interval. This suggests that the interval of convergence might be −1 < x < 1. To more
fully understand this power series, we apply the Ratio Test to determine the values of x
for which the power series converges absolutely. For the given series, we have
a k =
x k
k 2 ,
so
lim
k→∞
|a k+1 |
|a k |
= lim
k→∞
|x| k+1
(k+1) 2
|x| k
k 2
= lim
k→∞
|x|
k
k + 1
2
= |x| lim
k→∞
k
k + 1
2
= |x|.
Therefore, the Ratio Test tells us that the given power series f (x) converges absolutely
when |x| < 1 and diverges when |x| > 1. Since the Ratio Test is inconclusive when |x| = 1,
we need to check x = 1 and x = −1 individually.
When x = 1, observe that
f (1) =
∞
k=1
1
k 2 .