Matematicas
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 . . . . . . . . . . . . . . . . . . . . . 369 Series and Convergence . . . . . . . . . . . . . . 373 The Integral Test and p-Series Comparisons of Series . . . . . . . . . . 378
. . . . . . . . . . . . . . 381
Alternating Series . . . .. . . . . . . . . . . . . 385 The Ratio and Root Tests . . . . . . . . . . . . . 389 Taylor Polynomials and Approximations . . . . . 393 Power Series . . . . . . . . . . . . . . . . . . . . 398 Representation of Functions by Power Series . . 403
Section 8.10 Taylor and Maclaurin Series
. . . . . . . . . . . 408
Review Exercises . . . . . . . . . . . . . . . . . . . . . . . . . 414Problem Solving . . . . . . . . . . . . . . . . . . . . . . . . . 421
C H A P T E R Infinite Series
Section 8.1
8
Sequences
Solutions to Even-Numbered Exercises
2. an a1 a2 a3 a4 a5
2n n 2 4 4 5 6 6 8 7 10 8 5 4 2 n 1 3 1 2
4. an a1 a2 a3 a4 a5 4 9 2 3
2 3
n
6. an a1 a2
cos cos cos cos
n 2 2 0 1 3 2 0 1 0
8 27 16 81 32 243 2 n 2 1 2 3 1 2 2 5 6 n2 6 3 2 2 3 3 8 625 18 25 2 34 3 87 8 266 25
a3 a4 a5
cos 2 cos 5 2
8. an a1 a2 a3 a4 a5 2 5 2 3 2 1
1
n
1
10. an a1 a2
10 10 10 10 10 10
12. an a1 a2 a3 a4 a5
n
3n! 1! 3 6 9 12 15
3n
2 2 2 1
31 32 33 34 35
a3 a4 a5
2 4
1 2
14. a1 a2 a3 a4 a5
4, ak 1 2 2 2 3 2 4 2
k
1
1 2
ak
16. a1 a2 a3 a4 a5
6, ak 1 2 a 3 1 1 2 a 3 2 1 2 a 3 3 1 2 a 3 41
1 2 a 3 k 1 2 6 3 1 2 12 3 1 2 48 3 1 768 3
2
1 1 1 1
a1 a2 a3 a4
4 6 12 30
12 48 768 196,608
369
370
Chapter 8
Infinite Series
18. Because the sequence tends to 8 as n tends to infinity, it matches (a). 22. 24.
20. This sequence increases for a few terms, then decreases a2 16 8. Matches (b). 2 26.
4
4
10
−1
12 −1 −3 −1 12 −1 −1 12
an
2 n 25 2 6 2 2! n!
4 ,n n 6 6 6 11 2 6
1, . . . , 10
an
8 0.75
n
1,
n
1, 2, . . . , 10
an 25! 23!
3n2 ,n n2 1 23! 24 25 23! 24 25
1, . . . , 10
28. an a5 a6
30. an
1
2an, a1 2 40 2 80
5 80 160
32.
a5 a6
600
34.
n
n! n n 1 n2
1 n n! 1 n 2
2
36.
2n 2 ! 2n !
2n ! 2n 2n
1 2n 2n ! 2
2
1 2n
38. lim 5
n→
5
05
40. lim
n→
5n n2 4
n→
lim 5
5 1 4 n2
42. lim cos
n→
2 n
1
5 1 44.
2
46.
4
−1
12
−1 −1
12
−1
The graph seems to indicate that the sequence converges to 0. Analytically,
n→
The graph seems to indicate that the sequence converges to 3. Analytically,
n→
lim an
n→
lim
1 n3 2
x→
lim
1 x3 2
0.
lim an
n→
lim 31 2n
3
0
3.
3
48. lim 1
n→
1
n
50. lim
n→
n 1
3
n
1, converges
does not exist, (alternates between 0 and 2), diverges. 1 n2 1
n
52. lim
n→
0, converges
54. lim
n→
ln n n lim 1 2n
n→
lim
1 2 ln n n
n→
0, converges
(L’Hôpital’s Rule)
n
56. lim 0.5
n→
0, converges
58. lim
n→
n n!
2!
n→
lim
1 nn 10, converges
Section 8.1 n2 2n 1 n 1 x sin . x 1 x lim sin 1 x 1 x sin y y lim 1 x 2 cos 1 x 1 x2 1 n lim cos 1 x cos 0 1 (L’Hôpital’s Rule) 1 2n n2 1 2n2 4n2 1 1 , converges 2
Sequences
371
60. lim
n→
n→
lim
62. an
n sin
Let f x
x→
lim x sin
x→
x→
x→
or,
x→
lim
sin 1 x 1 x
y→0
lim
1. Therefore lim n sin
n→
1.
64. lim 21
n→n
20
1, converges
66. lim
n→
cos n n2
0, converges
68. an
4n
1
70. an
1n n2 2n
1
72. an
n 3n
2 1
74. an
1
n
3n 2n
2 1
76. an
1 2n
1
1 2n 1
78. an
1 n!
80. an
xn 1 n 1!
2n 3x x 2 6 x 2
82. Let f x
. Then f x
2.
84. an a1 a2 a3
ne
n 2
0.6065 0.7358 0.6694
Thus, f is increasing which implies...
Regístrate para leer el documento completo.