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

Section 11.10 Taylor and Maclaurin Series 771
1. If f sxd − o ǹ−0 bnsx 2 5dn for all x, write a formula for b 8.
2. The graph of f is shown.
(a) Explain why the series
y
1
0 1
x
1.6 2 0.8sx 2 1d 1 0.4sx 2 1d 2 2 0.1sx 2 1d 3 1 ∙ ∙ ∙
is not the Taylor series of f centered at 1.
(b) Explain why the series
2.8 1 0.5sx 2 2d 1 1.5sx 2 2d 2 2 0.1sx 2 2d 3 1 ∙ ∙ ∙
is not the Taylor series of f centered at 2.
3. If f snd s0d − sn 1 1d! for n − 0, 1, 2, . . . , find the Maclaurin
series for f and its radius of convergence.
4. Find the Taylor series for f centered at 4 if
f
f snd s4d − s21dn n!
3 n sn 1 1d
What is the radius of convergence of the Taylor series?
5–10 Use the definition of a Taylor series to find the first four
nonzero terms of the series for f sxd centered at the given value of a.
5. f sxd − xe x , a − 0 6. f sxd − 1
1 1 x , a − 2
7. f sxd − s 3 x , a − 8 8. f sxd − ln x, a − 1
9. f sxd − sin x, a − y6 10. f sxd − cos 2 x, a − 0
11–18 Find the Maclaurin series for f sxd using the definition of
a Maclaurin series. [Assume that f has a power series expan sion.
Do not show that R nsxd l 0.] Also find the associated radius of
convergence.
11. f sxd − s1 2 xd 22 12. f sxd − lns1 1 xd
13. f sxd − cos x 14. f sxd − e 22x
15. f sxd − 2 x 16. f sxd − x cos x
17. f sxd − sinh x 18. f sxd − cosh x
19–26 Find the Taylor series for f sxd centered at the given value
of a. [Assume that f has a power series expansion. Do not show
that R nsxd l 0.] Also find the associated radius of convergence.
19. f sxd − x 5 1 2x 3 1 x, a − 2
;
20. f sxd − x 6 2 x 4 1 2, a − 22
21. f sxd − ln x, a − 2 22. f sxd − 1yx, a − 23
23. f sxd − e 2x , a − 3 24. f sxd − cos x, a − y2
25. f sxd − sin x, a − 26. f sxd − sx , a − 16
27. Prove that the series obtained in Exercise 13 represents cos x
for all x.
28. Prove that the series obtained in Exercise 25 represents sin x
for all x.
29. Prove that the series obtained in Exercise 17 represents sinh x
for all x.
30. Prove that the series obtained in Exercise 18 represents cosh x
for all x.
31–34 Use the binomial series to expand the function as a power
series. State the radius of convergence.
31. s 4 1 2 x 32. s 3 8 1 x
33.
1
s2 1 xd 3
34. s1 2 xd3y4
35–44 Use a Maclaurin series in Table 1 to obtain the Maclaurin
series for the given function.
35. f sxd − arctansx 2 d 36. f sxd − sinsxy4d
37. f sxd − x cos 2x 38. f sxd − e 3x 2 e 2x
39. f sxd − x coss 1 2 x 2 d 40. f sxd − x 2 lns1 1 x 3 d
x
41. f sxd −
s4 1 x 2
x 2
42. f sxd −
s2 1 x
43. f sxd − sin 2 x fHint: Use sin 2 x − 1 2 s1 2 cos 2xd.g
44. f sxd −Hx 2 sin x
x 3 if x ± 0
1
6 if x − 0
45–48 Find the Maclaurin series of f (by any method) and its
radius of convergence. Graph f and its first few Taylor polynomials
on the same screen. What do you notice about the relation ship
between these polynomials and f ?
45. f sxd − cossx 2 d 46. f sxd − lns1 1 x 2 d
47. f sxd − xe 2x 48. f sxd − tan 21 sx 3 d
49. Use the Maclaurin series for cos x to compute cos 58 correct
to five decimal places.
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