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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 . . . . . . . . . . . . . . . . . . . . . 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...
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