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

The inequality | x 1 2 |
Section 11.8 Power Series 751
, 3 can be written as 25 , x , 1, so we test the series at
the endpoints 25 and 1. When x − 25, the series is
ns23d
ò
n
− 1
n−0 3 n11 3 ò s21d n n
n−0
which diverges by the Test for Divergence [s21d n n doesn’t converge to 0]. When
x − 1, the series is
ns3d
ò
n
− 1
n−0 3 n11 3 ò n
n−0
which also diverges by the Test for Divergence. Thus the series converges only when
25 , x , 1, so the interval of convergence is s25, 1d. n
1. What is a power series?
2. (a) What is the radius of convergence of a power series?
How do you find it?
(b) What is the interval of convergence of a power series?
How do you find it?
3–28 Find the radius of convergence and interval of convergence
of the series.
3. ò s21d n nx n
n−1
x
5. ò
n
n−1 2n 2 1
x
7. ò
n
n−0 n!
x
9. ò
n
n−1 n 4 4 n
s21d
4. ò
n x n
n−1 s 3 n
s21d
6. ò
n x n
n−1 n 2
8. ò n n x n
n−1
10. ò 2 n n 2 x n
n−1
s21d
11. ò
n 4 n
s21d
x n 12. ò
n21
x n
n−1 sn n−1 n5 n
n
13. ò
n−1 2 n sn 2 1 1d x x n 14. ò
2n
n−1 n!
sx 2 2d
15. ò
n
n−0 n 2 1 1
sx 1 2d
17. ò
n
n−2 2 n ln n
s21d
16. ò
n
n
sx 2 1dn
n−1 s2n 2 1d2
18. ò
n−1
sx 2 2d
19. ò
n
s2x 2 1d
20. ò
n
n−1 n n n−1
n
21. ò
n−1 b sx 2 n adn , b . 0
b
22. ò
n
n−2 ln n sx 2 adn , b . 0
n
23. ò n!s2x 2 1d n
24. ò
2 x n
n−1
n−1 2 ? 4 ? 6 ? ∙ ∙ ∙ ? s2nd
s5x 2 4d
25. ò
n
x
26. ò
2n
n−1 n 3 n−2 nsln nd 2
x
27. ò
n
n−1 1 ? 3 ? 5 ? ∙ ∙ ∙ ? s2n 2 1d
n!x
28. ò
n
n−1 1 ? 3 ? 5 ? ∙ ∙ ∙ ? s2n 2 1d
29. If o ǹ−0 cn4n is convergent, can we conclude that each of the
following series is convergent?
(a) ò c ns22d n
(b) ò c ns24d n
n−0
n−0
30. Suppose that o ǹ−0 cnxn converges when x − 24 and diverges
when x − 6. What can be said about the convergence or divergence
of the following series?
(a) ò c n
(b) ò c n8 n
n−0
n−0
(c) ò c ns23d n
(d) ò s21d n c n 9 n
n−0
n−0
sn
n
sx 1 6dn
8 31. If k is a positive integer, find the radius of convergence of
the series
5 n sn
ò
n−0
sn!d k
sknd! x n
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