Xxxxxxxx
Section 8.1 Section 8.2 Section 8.3 Section 8.4 Section 8.5 Section 8.6 Section 8.7 Section 8.8 Section 8.9
8
Sequences . . . . . . . . . . . . . . . . . . . . . 121 Series and Convergence . . . . . . . . . . . . . . 126 The Integral Test and p-Series Comparisons of Series . . . . . . . . . . 131
. . . . . . . . . . . . . . 135
Alternating Series . . . .. . . . . . . . . . . . . 138 The Ratio and Root Tests . . . . . . . . . . . . . 142 Taylor Polynomials and Approximations . . . . . 147 Power Series . . . . . . . . . . . . . . . . . . . . 152 Representation of Functions by Power Series . . 157
Section 8.10 Taylor and Maclaurin Series
. . . . . . . . . . . 160
Review Exercises . . . . . . . . . . . . . . . . . . . . . . . . . 167Problem Solving . . . . . . . . . . . . . . . . . . . . . . . . . 172
C H A P T E R Infinite Series
Section 8.1
8
Sequences
Solutions to Odd-Numbered Exercises
2n 21 2
2
1. an a1 a2 a3 a4 a5
3. an 2 4 8 16 32 a3 a4 a5 a1 a2
1 2 1 2 1 2 1 2 1 2 1 2
n
5. an
1
sin sin sin sin
n 2 2 1 0 3 2 0 1 1
1 2 1 4 1 8 1 16 1 32 1 n2 1 1 4 1 9 1 16 1 25 5 19 4 43 9 77 16 121 25a1 a2 a3 a4 a5
23 2 2
4 5
2
3
sin 2 sin 5 2
4
5
7. an a1 a2 a3 a4 a5
1 1 12 1 22 1 32
nn
1 2
n2
1
9. an 1 a1 a2 a3 a4 a5
5 5 5 5 5 5
1 n 1 1 2 1 3 1 4 1 5
11. an a1 a2 a3 a4 a5
3n n! 3 1! 32 2! 33 3! 34 4! 35 5! 3 9 2 27 6 81 24 243 120
3
1 4 1 9 1 16 1 25
6
1 10 42 1 15 52
13. a1 a2
3, ak 2 a1 23
1
2 ak 1
1
15. a1a2 a3
32, ak 1 a 2 1 1 a 2 2 1 a 2 3 1 a 2 4
1
1 a 2 k 16 8 4 2
1 1 1 1 1 1 1
4
1 32 2 1 16 2 1 8 2 1 4 2
a3
2 a2 24
6 a4 10 a5 18
a4
2 a3 26
a5
2 a4 2 10
121
122
Chapter 8
Infinite Series 8 2 1
8 3,
17. Because a1 8 1 1 4 and a2 the sequence matches graph (d). 21.
8
19. This sequence decreases and a1 Matches (c).
4, a2
4 0.5
2.23.
18
25.
3
−1 −1 −1 12 − 10
12
−1 −1
12
an
2 n, n 3
1, . . . , 10
an
16
0.5
n
1,
n
1, . . . , 10
an
2n n 1
,n
1, 2, . . . , 10
27. an a5 a6
3n 35 36
1 1 1 14 17
29. an an a6
3 2n 3 2 3 2
4
1
31. 3 16 3 32
1 2.
10! 8!
8! 9 10 8! 9 10 90
Add 3 to preceeding term.
5
Multiply the preceeding termby n n! 1! n! n 1 n! n 1 2n 2n 1! 1!
33.
35.
2n 1 2n 2n
2n 1 ! 1 ! 2n 2n 1 1 n
1
37. lim
n→
5n2 n2 2
5
39. lim
n→
2n n2 1
n→
lim 2
2 1 1 n2
41. lim sin
n→
0
2 1 43.
3
45.
−1
2
12
−1 −1
12 −2
The graph seems to indicate that the sequence converges to 1. Analytically,
n→
The graph seems to indicate that the sequence diverges.Analytically, the sequence is an
n→
lim an
n→
lim
n n
1
x→
lim
x x
1
x→
lim 1
1.
0,
1, 0, 1, 0,
1, . . . .
Hence, lim an does not exist. 3n2 n 4 2n2 1 3 , converges 2
47. lim
n→
n n 1 does not exist (oscillates between 1
n
49. lim
n→
1 and 1), diverges. ln n3 2n 3 ln n 2 n 3 1 2 n 0, converges
51. lim
n→
1 n
1
n
0,converges
53. lim
n→
n→
lim
n→
lim
(L’Hôpital’s Rule)
Section 8.1 3 4 n n
n
Sequences
123
55. lim
n→
0, converges
57. lim
n→
n n!
1!
n→
lim n
1
, diverges
59. lim
n→
1 n
n 1
n→
lim
n 1 n2
1 nn 2n n
2
n2 1 0, converges
61. lim
np 0, converges n→ en p > 0, n ≥ 2
n→
lim
63. an
1 lim 1
k n
n65. lim
n→
sin n n
n→
lim sin n
1 n
0, converges
n→
k n
n u→0
lim 1
u
1 u k
ek
where u
k , converges n 2 69. an n2 2 71. an n n 1 2 n 1 n
67. an
3n
73. an
1 2n
n 2
1
75. an
1
1 n
n n
1
77. an
n
2
79. an
1
3
1n 1 5 . . . 2n
1 1
n 1 2nn! 2n !
81. an
1 1 < 4 an n n 1 monotonic; an < 4...
Regístrate para leer el documento completo.