Math 152 Class Notes
November 12, 2015
10.6 Representations of Functions as Power Series
In this section we learn how to represent certain types of functions as sums of power
series by manipulating geometric series or by dierentiating or integrating such a series.
Example 1.
∞
P
1
=
xn = 1 + x + x2 + x3 + · · ·
1 − x n=0
Example 2. Find a power series for
3
and nd the interval of convergence.
2x − 1
Example 3. Find a power series for
1
and nd the interval of convergence.
1 + 2x3
Example 4. Find a power series for
x
and nd the interval of convergence.
3−x
Dierentiation and Integration of Power Series
If the power series
f dened by
∞
P
n=0
cn (x − a)n has radius of convergence R > 0, then the function
2
3
f (x) = c0 + c1 (x − a) + c2 (x − a) + c3 (x − a) + · · · =
∞
X
cn (x − a)n
n=0
is dierentiable (and therefore continuous) on the interval (a − R, a + R) and
(a) f (x) = c1 + 2c2 (x − a) + 3c3 (x − a) + · · · =
0
(b)
ˆ
2
∞
P
ncn (x − a)n−1
n=1
∞
X
(x − a)3
(x − a)n+1
(x − a)2
+ c2
+ ··· = C +
cn
f (x)dx = C + c0 (x − a) + c1
2
3
n+1
n=0
The radii of convergence of the power series in Equations (a) and (b) are both R.
This rule says that we can dierentiate and integrate a power series term-by-term just
as we do for a polynomial.
Example 5. f (x) =
∞ (4x)n
P
n=0 n!
(a) Find f 0 (x).
(b) Find
(c) Find
ˆ
ˆ
0
f (x)dx.
1
f (x)dx.
Example 6. Find a power series for
1
and nd the interval of convergence.
(1 + x)2
x2
and nd the interval of convergence.
Example 7. Find a power series for
(1 − 3x)2
Example 8. (a) Find a power series for ln(1 + x) and nd the interval of convergence.
(b) Find the sum of
∞
P
(−1)n−1
n=1
1
1 1 1
= 1 − + − + ···
n
2 3 4
Example 9. Find a power series for ln(3 + x).
Example 10. (a) Find a power series for arctan x and nd the interval of convergence.
(b) Find the sum of
Example 11. Find
ˆ
∞
P
(−1)n
n=0
1
1 1 1
= 1 − + − + ···
2n + 1
3 5 7
arctan x
dx as a power series.
x
Example 12. (a) Evaluate
ˆ
0
(b) Approximate
ˆ
0
0.1
0.1
1
dx as a series.
1 + x3
1
dx correct to within 0.001.
1 + x3