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C H A P T E R Infinite Series
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...
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