James Stewart-Calculus_ Early Transcendentals-Cengage Learning (2015)

Section 11.9 Representations of Functions as Power Series 757
(b) In applying the Fundamental Theorem of Calculus, it doesn’t matter which antiderivative
we use, so let’s use the antiderivative from part (a) with C − 0:
y 0.5
0
1
1 1 x dx − Fx 2 x 8
7 8 1 x 15
15 2 x 22
1y2
22 1 ∙ ∙ ∙ G0
− 1 2 2 1
8 ∙ 2 8 1 1
15 ∙ 2 15 2 1
22 ∙ 2 22 1 ∙ ∙ ∙ 1 s21d n
s7n 1 1d2 7n11
1 ∙ ∙ ∙
This infinite series is the exact value of the definite integral, but since it is an alternating
series, we can approximate the sum using the Alternating Series Estimation Theorem.
If we stop adding after the term with n − 3, the error is smaller than the term with
n − 4:
1
29
< 6.4 3 10211
29 ∙ 2
So we have
y 0.5
0
1
1 1 x dx < 1 7 2 2 1
8 ? 2 1 1
8 15 ? 2 2 1
15 22
< 0.49951374 n
22 ? 2
1. If the radius of convergence of the power series o ǹ−0 cnxn
is 10, what is the radius of convergence of the series
o ǹ−1 ncnxn21 ? Why?
2. Suppose you know that the series o ǹ−0 bnxn converges for
| x | , 2. What can you say about the following series? Why?
ò
n−0
b n
n 1 1 x n11
3–10 Find a power series representation for the function and
determine the interval of convergence.
3. f sxd − 1
1 1 x
5. f sxd − 2
3 2 x
7. f sxd − x 2
x 4 1 16
9. f sxd − x 2 1
x 1 2
4. f sxd −
5
1 2 4x 2
6. f sxd − 4
2x 1 3
8. f sxd −
x
2x 2 1 1
10. f sxd − x 1 a
x 2 1 a 2 , a . 0
11–12 Express the function as the sum of a power series by first
using partial fractions. Find the interval of convergence.
11. f sxd − 2x 2 4
x 2 2 4x 1 3
12. f sxd − 2x 1 3
x 2 1 3x 1 2
13. (a) Use differentiation to find a power series representation for
1
f sxd −
s1 1 xd 2
What is the radius of convergence?
(b) Use part (a) to find a power series for
1
f sxd −
s1 1 xd 3
(c) Use part (b) to find a power series for
x 2
f sxd −
s1 1 xd 3
14. (a) Use Equation 1 to find a power series representation for
f sxd − lns1 2 xd. What is the radius of convergence?
(b) Use part (a) to find a power series for f sxd − x lns1 2 xd.
(c) By putting x − 1 2 in your result from part (a), express ln 2
as the sum of an infinite series.
15–20 Find a power series representation for the function and
determine the radius of convergence.
15. f sxd − lns5 2 xd 16. f sxd − x 2 tan 21 sx 3 d
17. f sxd −
x
s1 1 4xd 2
18. f sxd − S x
3
2 2 xD
19. f sxd − 1 1 x
s1 2 xd 2 20. f sxd − x 2 1 x
s1 2 xd 3
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