1
10.6: Functions as Power Series
As stated previously, the power series
∞
X
cn (x − a)n represents a function whose domain is the interval
n=0
of convergence of the series. In this section, we use derivatives and integrals to obtain new power series.
Recall:
d
(f (x) + g(x)) =
dx
ˆ
(f (x) + g(x)) dx =
Claim: If
cn (x − a)n has radius of convergence r, then
n=0
r.
If
∞
X
∞
X
cn n(x − a)n−1 has radius of convergence
n=0
cn (x − a)n converges to f (x), what should
n=0
∞
X
cn n(x − a)n−1 converge to?
n=0
ˆ
In like manner,
∞
X
∞
X
!
cn (x − a)n
dx has radius of convergence r and converges to
n=0
Examples: Given f (x) =
ˆ
∞
X
xn
, find a power series for f 0 (x) and f (x) dx. What is f (x)?
n!
n=0
1
Find a power series for f (x) =
1
and use it to determine a power series for f (x) = arctan x.
1 + x2
.
Find the radius and interval of convergence of the series.
π
Choosing an appropriate value of x, how many terms of the series are needed to approximate
to
4
within 0.001?
2
On Beyond Average:
Find the sum of
∞
X
nxn .
n=0
If g(x) =
(−1)n n
1 2 1 3 1 4
x − x + x − ··· +
x + · · · , find g 0
2
3
4
n
3
1
2