Chapter 8

Section 8.1

1. False

2. False

3. True

4. (b)

5. False

6. True

7. False

8. False

9. False

10. (b)

11. True

12. False

13.

15. 0, ln 2, ln 3, ln 4

17.

19. 1,−1, 1,−1

21.

23. a_n = 2n

25. a_n = 2ⁿ

27. a_n =

29. a_n =

31. a_n = (n − 1)!

33. 0

35. 1

37. 4

39. 0

41. 0

43. 1

45.

47. e⁻²

49. 0

51. 1

53.

55. 1

57. 0

59. 0

61. 0

63. Diverges

65. Diverges

67. Converges

69. Diverges

71. Converges

73. Bounded from above and from below

75. Bounded from below

77. Bounded from below

79. Bounded from above and from below

81. Nonmonotonic

83. Nonmonotonic

85. Monotonic decreasing

87. Nonmonotonic

89. Decreasing, bounded from below

91. Increasing, bounded from above

93. Increasing, bounded from above

95. Converges; 6

97. Converges; −ln 3

99. Diverges

101. Converges; 0

103. Converges; 0

105. Converges; 0

107. Converges; 1

109. Converges; 1

111. Converges; 1

113. Converges; 1

115. Converges

117. Converges

119. Converges

121. Diverges

123. Converges

125. Converges

127. p_n = 3000rⁿ + Converges

129. (a) I_n = 0.95ⁿ I₀

(b) 77

131. (a) See Student Solutions Manual.

(b)

(c)

133. See Student Solutions Manual.

135. See Student Solutions Manual.

137. See Student Solutions Manual.

139. Answers will vary.

141. See Student Solutions Manual.

143. Diverges

145. See Student Solutions Manual.

147. See Student Solutions Manual.

149. See Student Solutions Manual.

151. See Student Solutions Manual.

153. See Student Solutions Manual.

Section 8.2

1. (1)

2. (d)

3. True

4. False

5.

6. False

7.

9. 10

11.

13. −

15.

17. Diverges

19. Converges; 6

21. Converges;

23. Converges;

25. Converges; 6

27. Converges;

29. Diverges

31. Diverges

33. Converges;

35. Diverges

37. Converges;

39. Diverges

41. Converges;

43. Diverges

45. Converges;

47. Diverges

49. Diverges

51. Diverges

53. Converges;

55. Diverges

57. Converges; sin 1

59.

61.

63. 90 ft

65. (a)

(b) h = 2000

67. (a)

(b) $6.67

69. See Student Solutions Manual.

71. (a)

(b)

(c) e^min =

(d) kg

73. See Student Solutions Manual.

75. n = 11

77. See Student Solutions Manual.

79. See Student Solutions Manual.

81. See Student Solutions Manual.

83. See Student Solutions Manual.

85. See Student Solutions Manual.

87. See Student Solutions Manual.

Section 8.3

1. (a) 0

2. False

3. True

4. True

5. False

6. True

7. True

8. False

9. p > 1; 0 < p ≤ 1

10. Converges

11. Diverges

12. False

13. Diverges

15. Diverges

17. Diverges

19. Converges

21. Diverges

23. Converges

25. Converges

27. Diverges

29. Converges

31. Diverges

33. Converges

35. Converges

37. Converges

39. Diverges

41. Diverges

43. Diverges

45. Diverges

47. Diverges

49. Diverges

51. Diverges

53. Diverges

55. See Student Solutions Manual.

57. See Student Solutions Manual.

59. (a)

(b) \(\pi^{2}x\,9.8099\)

61. Converges

63. Converges

65. See Student Solutions Manual.

67. See Student Solutions Manual.

69. See Student Solutions Manual.

71. See Student Solutions Manual.

73. (−∞, −1)

75. (a) 1.1975

(b) Upper is 1.5, lower is 0.5.

(c) See Student Solutions Manual.

Section 8.4

1. (b)

2. False

3. False

4. False

5. Converges

7. Converges

9. Diverges

11. Converges

13. Converges

15. Converges

17. Diverges

19. Converges

21. Converges

23. Converges

25. Converges

27. Converges

29. Diverges

31. Converges

33. Diverges

35. Converges

37. Diverges

39. Converges

41. Converges

43. Diverges

45. Diverges

47. Converges

49. See Student Solutions Manual.

51. See Student Solutions Manual.

53. Diverges

55. Converges

57. Answers will vary.

59. (a) See Student Solutions Manual.

(b) See Student Solutions Manual.

61. Diverges

63. Converges

65. See Student Solutions Manual.

67. Answers will vary; for example: ; diverges converges

69. Diverges

71. See Student Solutions Manual.

Section 8.5

1. False

2. False

3. False

4. False

5. False

6. True

7. Converges

9. Diverges

11. Diverges

13. Converges

15. Diverges

17. Diverges

19. 0.8611; upper estimate to the error is 0.0625

21. 0.9498; upper estimate to the error is 0.0039

23. 0.7222; upper estimate to the error is 0.00617

25. 0.3218; upper estimate to the error is 6.53 × 10^-5

27. Conditionally convergent

29. Absolutely convergent

31. Absolutely convergent

33. Diverges

35. Absolutely convergent

37. Absolutely convergent

39. Conditionally convergent

41. Absolutely convergent

43. Conditionally convergent

45. Absolutely convergent

47. Diverges

49. Absolutely convergent

51. See Student Solutions Manual.

53. See Student Solutions Manual.

55. See Student Solutions Manual.

57. Absolutely convergent for |r| < 1; divergent if |r| ≥ 1

59. See Student Solutions Manual.

61. See Student Solutions Manual.

63. See Student Solutions Manual.

65. Absolutely convergent

67. See Student Solutions Manual.

69. Absolutely convergent if 2 < p < 3; divergent if p ≥ 3

71. Absolutely convergent

73. Converges

75. See Student Solutions Manual.

Section 8.6

1. False

2. False

3. False

4. False

5. Converges

7. Converges

9. Converges

11. Converges

13. Diverges

15. Converges

17. Diverges

19. Converges

21. Diverges

23. Converges

25. Diverges

27. Converges

29. Converges

31. Converges

33. Converges

35. Converges

37. Converges

39. Converges

41. Converges

43. Converges

45. See Student Solutions Manual.

47. Answers will vary.

49. (a) Converges by the ratio test

(b)

51. See Student Solutions Manual.

53. Converges if |x| ≤ 1; diverges if |x| > 1

55. See Student Solutions Manual.

57. See Student Solutions Manual.

59. See Student Solutions Manual.

Section 8.7

1. False

2. False

3. True

4. False

5. False

6. ak, bk

7. Diverges by theTest for Divergence

9. Absolutely convergent by the Geometric Series Test

11. Absolutely convergent by the Limit Comparison Test

13. Diverges by the Comparison Test for Divergence

15. Diverges by the Ratio Test

17. Diverges by the Test for Divergence

19. Conditionally convergent by the Alternating Series Test

21. Diverges by the Ratio Test

23. Absolutely convergent by the Ratio Test

25. Diverges by the Limit Comparison Test

27. Absolutely convergent by the Comparison Test for Convergence

29. Diverges by the Test for Divergence

31. Absolutely convergent by the Root Test

33. Absolutely convergent by the Root Test

35. Absoutely convergent by the Integral Test

37. Diverges by the Root Test

39. Absolutely convergent by the Comparison Test for Convergence

41. Converges;

43. Converges by the Ratio Test.

45. (a) See Student Solutions Manual.

(b)

47. Converges

Section 8.8

1. True

2. True

3. True

4. False

5. True

6. False

7. False

8. True

9. True

10. False

11. False

12. False

13. −1 < x < 1

15. −4 < x < 2

17. (a, b) R = 2; −2 ≤ x < 2

(c) Answers will vary.

19. (a, b) R = 1; −1 ≤ x < 1

(c) Answers will vary.

21. (a, b) R = 3; −3 < x < 3

(c) Answers will vary.

23. (a, b) R = 1; −1 < x < 1

(c) Answers will vary.

25. (a, b) R = 1; 2 < x < 4

(c) Answers will vary.

27. R = 1; −1 ≤ x ≤ 1

29. R = 1; 1 ≤ x ≤ 3

31. R = ∞; −∞ < x < ∞

33. R = 1; −1 < x < 1

35. R = 4; −4 < x < 4

37. R = 3; 0 < x < 6

39. R = ∞; −∞ < x < ∞

41. R = ∞; −∞ < x < ∞

43.

45. (a) −3 < x < 3

(b) f (2) = 3, f (−1) =

(c) f (x) =

47. (a) 0 < x < 4

(b) f (1) = , f (2) = 1

(c) f (x) =

49. Converges at x = 2; no information about x = 5

51. (a) True

(b) False

(c) False

(d) False

(e) True

(f) True

53. (a) f (x) =

(b) R = 1; −1 < x < 1

55. (a) f (x) =

(b) R = 3; −3 < x < 3

57. (a) f (x) =

(b) R = 1; −1 < x < 1

59. (a) f (x) =

(b) \(\int{f(x)}\,dx=C+\sum\limits_{k=0}^{\infty}{\dfrac{(-1)^{k}x^{2k+2}}{(2k+2)!}}\)

61. (a)

(b)

63. \(f(x)=\sum\limits_{k=1}^{\infty}{(-1)^{k-1}kx^{k-1}}\); −1 < x < 1

65. f (x) = ; −1 < x < 1

67. f (x) = ; −1 < x ≤ 1

69. f (x) = ; −1 < x < 1

71. −1 ≤ x < 1

73. −1 ≤ x < 1

75. −1 < x < 1

77. −3 < x < 5

79. (a)

(b) −1 < x < 1

81. 0.693

83. See Student Solutions Manual.

85. See Student Solutions Manual.

87. See Student Solutions Manual.

89. −1 ≤ x < 5

91. ; R = ∞

Section 8.9

1. Taylor series

2. Maclaurin

3. f (x) =

5. f (x) =

7. f (x) =

9. f (x) =

11. f (x) =

13. f (x) =

15. f (x) =

17. f (x) =

19. f (x) =

21. f (x) =

23. f (x) = −6 + 5x + 2x² + 3x³

25. f (x) = 4 + 18(x − 1) + 11(x − 1)² + 3(x − 1)³

27. f (x) =

29. x + x²

31. x − x²

33. x +

35.

37. ; interval: [−1, 1]

39. ; interval: (−1, 1]

41. ; interval: (−1, 1]

43.

45. cos x =

47. \(\sec{x}= 1+\dfrac{x^{2}}{2}+\dfrac{5x^{4}}{24}+\dfrac{61x^{6}}{720} +\dfrac{277x^{8}}{8064} + \ldots\)

49. \(C + \sum\limits_{k=0}^{\infty}{\dfrac{(-1)^{k}x^{2k+1}}{2k+1}}\)

51.

53. See Student Solutions Manual.

55. See Student Solutions Manual.

57. See Student Solutions Manual.

59.

61. See Student Solutions Manual.

63. See Student Solutions Manual.

65. + ...

Section 8.10

1. (a)

(b)

(c) cos(π/90) ≈ 0.999

(d) |E| ≤ 5.47 × 10⁻¹⁷

(e) |x| ≤ 1.1904

3. (a)

(b) [−1, 1]

(c)

(d)

(e) Answers will vary.

(f) ≈0.965; the error is ≤ 3.018 × 10⁻⁷

5. (a)

(b) [−1, 1]

(c) x −

(d)

(e) Answers will vary.

7. 0.310

9. 0.195

11. 0.200

13. 0.487

15. See Student Solutions Manual.

17. (a) 0.0698

(b) 0.9659

(c) 0.04996

19. 1538665/489888 ≈ 3.14085

21. (a)

(b) See Student Solutions Manual

(c) See Student Solutions Manual

23. (a) See Student Solutions Manual.

(b) 1.571

(c) 0.524

Review Exercises

1.

3. a_n =

5. 0

7. 0

9. Diverges

11. Converges; 0

13. See Student Solutions Manual.

15.

17. Converges;

19. Diverges

21. See Student Solutions Manual.

23. Converges

25. Converges

27. Converges

29. Converges

31. Converges; 1.079

33. Converges;−1.715

35. Conditionally convergent

37. Conditionally convergent

39. Converges

41. Diverges

43. Diverges

45. Converges

47. Diverges

49. Converges

51. Converges

53. (a) R = 1

(b) 2 ≤ x ≤ 4

55. (a) R = ∞

(b) all x

57. (a) R = 1

(b) 0 ≤ x < 2

59. f (x) = , −3 < x < 3

61. (a)

(b) 0.758

63.

65.

67.

69.

71. 1.349

73. 0.545