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

772 Chapter 11 Infinite Sequences and Series
50. Use the Maclaurin series for e x to calculate 1ys 10 e correct to
five decimal places.
51. (a) Use the binomial series to expand 1ys1 2 x 2 .
(b) Use part (a) to find the Maclaurin series for sin 21 x.
52. (a) Expand 1ys 4 1 1 x as a power series.
(b) Use part (a) to estimate 1ys 4 1.1 correct to three decimal
places.
53–56 Evaluate the indefinite integral as an infinite series.
53. y s1 1 x 3 dx 54. y x 2 sinsx 2 d dx
55. y cos x 2 1
x
dx
56. y arctansx 2 d dx
57–60 Use series to approximate the definite integral to within
the indicated accuracy.
57. y 1y2
x 3 arctan x dx (four decimal places)
0
58. y 1
sinsx 4 d dx (four decimal places)
0
59. y 0.4
0
60. y 0.5
0
s1 1 x 4 dx s| error | , 5 3 10 26 d
x 2 e 2x 2 dx s| error | , 0.001d
61–65 Use series to evaluate the limit.
x 2 lns1 1 xd
61. lim
62. lim
x l 0 x 2 x l 0
sin x 2 x 1 1 6
63. lim
x 3
x l 0 x 5
s1 1 x 2 1 2 1 2
64. lim
x
x l 0 x 2
65. lim
x l 0
x 3 2 3x 1 3 tan 21 x
x 5
66. Use the series in Example 13(b) to evaluate
tan x 2 x
lim
x l 0 x 3
1 2 cos x
1 1 x 2 e x
We found this limit in Example 4.4.4 using l’Hospital’s Rule
three times. Which method do you prefer?
67–72 Use multiplication or division of power series to find the
first three nonzero terms in the Maclaurin series for each function.
67. y − e 2x2 cos x 68. y − sec x
69. y − x
sin x
70. y − e x lns1 1 xd
71. y − sarctan xd 2 72. y − e x sin 2 x
;
73–80 Find the sum of the series.
73. òs21d x 4n
n
n−0 n!
3n
n21
75. ò s21d
n−1 n 5 n
s21d
77. ò
n 2n11
n−0 4 2n11 s2n 1 1d!
78. 1 2 ln 2 1
sln 2d2
2!
2
sln 2d3
3!
79. 3 1 9 2! 1 27
3! 1 81
4! 1 ∙ ∙ ∙
80.
s21d
74. ò
n 2n
n−0 6 2n s2nd!
76. 3
ò
n
n−0 5 n n!
1 ∙ ∙ ∙
1
1 ? 2 2 1
3 ? 2 1 1
3 ? 5 ? 2 2 1
5 7 ? 2 1 ∙ ∙ ∙
7
81. Show that if p is an nth-degree polynomial, then
psx 1 1d − o n
i−0
p sid sxd
i!
82. If f sxd − s1 1 x 3 d 30 , what is f s58d s0d?
83. Prove Taylor’s Inequality for n − 2, that is, prove that if
| f -sxd | < M for | x 2 a | < d, then
| R2sxd | < M 6 | x 2 a | 3 for | x 2 a | < d
84. (a) Show that the function defined by
f sxd −H e21yx 2
0
if x ± 0
if x − 0
is not equal to its Maclaurin series.
(b) Graph the function in part (a) and comment on its behavior
near the origin.
85. Use the following steps to prove (17).
(a) Let tsxd − o ǹ−0 (nk )x n . Differentiate this series to show
that
t9sxd − ktsxd
1 1 x
21 , x , 1
(b) Let hsxd − s1 1 xd 2k tsxd and show that h9sxd − 0.
(c) Deduce that tsxd − s1 1 xd k .
86. In Exercise 10.2.53 it was shown that the length of the ellipse
x − a sin , y − b cos , where a . b . 0, is
L − 4a y y2
s1 2 e 2 sin 2 d
0
where e − sa 2 2 b 2 ya is the eccentricity of the ellipse.
Expand the integrand as a binomial series and use the
result of Exercise 7.1.50 to express L as a series in powers of
the eccentricity up to the term in e 6 .
Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).
Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.