8.6. POWER SERIES
523
This is a p-series with p > 1, which we know converges. When x = −1, we have
f (−1) =
∞
k=1
(−1) k
k 2 .
This is an alternating series, and since the sequence
1
n 2
decreases to 0, the power series
converges when x = −1 by the Alternating Series Test. Thus, the interval of convergence
of this power series is −1 ≤ x ≤ 1.
Activity 8.28.
Determine the interval of convergence of each power series.
(a)
∞
k=1
(x − 1) k
3k
(b)
∞
k=1
k x
k
(c)
∞
k=1
k 2 (x + 1) k
4 k
(d)
∞
k=1
x k
(2k)!
(e)
∞
k=1
k!x
k
⊳
Manipulating Power Series
Recall that we know several power series expressions for important functions such as
sin(x) and e x . Often, we can take a known power series expression for such a function
and use that series expansion to find a power series for a different, but related, function.
The next activity demonstrates one way to do this.
Activity 8.29.
Our goal in this activity is to find a power series expansion for f (x) =
1
1 + x 2 centered
at x = 0.
523
This is a p-series with p > 1, which we know converges. When x = −1, we have
f (−1) =
∞
k=1
(−1) k
k 2 .
This is an alternating series, and since the sequence
1
n 2
decreases to 0, the power series
converges when x = −1 by the Alternating Series Test. Thus, the interval of convergence
of this power series is −1 ≤ x ≤ 1.
Activity 8.28.
Determine the interval of convergence of each power series.
(a)
∞
k=1
(x − 1) k
3k
(b)
∞
k=1
k x
k
(c)
∞
k=1
k 2 (x + 1) k
4 k
(d)
∞
k=1
x k
(2k)!
(e)
∞
k=1
k!x
k
⊳
Manipulating Power Series
Recall that we know several power series expressions for important functions such as
sin(x) and e x . Often, we can take a known power series expression for such a function
and use that series expansion to find a power series for a different, but related, function.
The next activity demonstrates one way to do this.
Activity 8.29.
Our goal in this activity is to find a power series expansion for f (x) =
1
1 + x 2 centered
at x = 0.