1. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 9- 1

2. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Chapter 9 Infinite Series

3. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall 9.1 Power Series

4. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 9- 4 Quick Review

5. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 9- 5 Quick Review

6. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 9- 6 Quick Review Solutions

7. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 9- 7 Quick Review Solutions

8. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 9- 8 What you’ll learn about Geometric Series Representing Functions by Series Differentiation and Integration Identifying a Series … and why Power series are important in understanding the physical universe and can be used to represent functions.

9. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 9- 9 Infinite Series

10. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 9- 10 Example Identifying a Divergent Series

11. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 9- 11 Example Identifying a Convergent Series

12. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 9- 12 Infinite Series

13. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 9- 13 Geometric Series

14. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 9- 14 Example Analyzing Geometric Series

15. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 9- 15 Example Analyzing Geometric Series

16. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 9- 16 Power Series

17. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 9- 17 Example Finding a Power Series by Differentiation

18. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 9- 18 Term-by-Term Differentiation

19. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 9- 19 Example Finding a Power Series by Integration

20. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 9- 20 Term-by-Term Integration

21. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall 9.2 Taylor Series

22. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 9- 22 Quick Review

23. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 9- 23 Quick Review Solutions

24. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 9- 24 What you’ll learn about Constructing a Series Series for sin x and cos x Beauty Bare Maclaurin and Taylor Series Combining Taylor Series Table of Maclaurin Series … and why The partial sums of a Taylor series are polynomials that can be used to approximate the function represented by the series.

25. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 9- 25 Example Constructing a Power Series for sin x

26. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 9- 26 Taylor Series Generated by f at x=0 (Maclaurin Series)

27. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 9- 27 Example Approximating a Function near 0

28. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 9- 28 Taylor Series Generated by f at x=a

29. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 9- 29 Example A Taylor Series at x = 1

30. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 9- 30 Example A Taylor Polynomial for a Polynomial

31. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 9- 31 Example A Taylor Polynomial for a Polynomial

32. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 9- 32 Maclaurin Series

33. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 9- 33 Maclaurin Series

34. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall 9.3 Taylor’s Theorem

35. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 9- 35 Quick Review

36. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 9- 36 Quick Review

37. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 9- 37 Quick Review Solutions

38. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 9- 38 Quick Review Solutions

39. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 9- 39 What you’ll learn about Taylor Polynomials The Remainder Remainder Estimation Theorem Euler’s Formula … and why If we approximate a function represented by a power series by its Taylor polynomials, it is important to know how to determine the error in the approximation.

40. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 9- 40 Example Approximating a Function to Specifications

41. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 9- 41 Taylor’s Theorem with Remainder

42. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 9- 42 Example Proving Convergence of a Maclaurin Series

43. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 9- 43 Remainder Estimation Theorem

44. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 9- 44 Example Proving Convergence

45. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 9- 45 Euler’s Formula

46. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 9- 46 Quick Quiz Sections 9.1-9.3

47. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 9- 47 Quick Quiz Sections 9.1-9.3

48. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 9- 48 Quick Quiz Sections 9.1-9.3

49. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 9- 49 Quick Quiz Sections 9.1-9.3

50. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 9- 50 Quick Quiz Sections 9.1-9.3

51. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 9- 51 Quick Quiz Sections 9.1-9.3

52. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall 9.4 Radius of Convergence

53. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 9- 53 Quick Review

54. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 9- 54 Quick Review

55. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 9- 55 Quick Review Solutions

56. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 9- 56 Quick Review Solutions

57. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 9- 57 What you’ll learn about Convergence nth-Term Test Comparing Nonnegative Series Ratio Test Endpoint Convergence … and why It is important to develop a strategy for finding the interval of convergence of a power series and to obtain some tests that can be used to determine convergence of a series.

58. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 9- 58 The Convergence Theorem for Power Series

59. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 9- 59 The nth-Term Test for Divergence

60. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 9- 60 The Direct Comparison Test

61. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 9- 61 Example Proving Convergence by Comparison

62. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 9- 62 Absolute Convergence

63. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 9- 63 Absolute Convergence Implies Convergence

64. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 9- 64 Example Using Absolute Convergence

65. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 9- 65 The Ratio Test

66. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 9- 66 Example Finding the Radius of Convergence

67. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 9- 67 Example Determining Convergence of a Series

68. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall 9.5 Testing Convergence at Endpoints

69. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 9- 69 Quick Review

70. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 9- 70 Quick Review

71. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 9- 71 Quick Review Solutions

72. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 9- 72 Quick Review Solutions

73. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 9- 73 What you’ll learn about Integral Test Harmonic Series and p-series Comparison Tests Alternating Series Absolute and Conditional Convergence Intervals of Convergence A Word of Caution … and why Additional tests for convergence of series are introduced in this section.

74. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 9- 74 The Integral Test

75. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 9- 75 Example Applying the Integral Test

76. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 9- 76 Harmonic Series p-series

77. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 9- 77 The Limit Comparison Test (LCT)

78. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 9- 78 Example Using the Limit Comparison Test

79. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 9- 79 The Alternating Series Test (Leibniz’s Theorem)

80. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 9- 80 The Alternating Series Estimation Theorem

81. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 9- 81 Rearrangement of Absolutely Convergent Series

82. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 9- 82 Rearrangement of Conditionally Convergent Series

83. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 9- 83 How to Test a Power Series for Convergence

84. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 9- 84 Example Finding Intervals of Convergence

85. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 9- 85 Procedure for Determining Convergence

86. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 9- 86 Quick Quiz Sections 9.4 and 9.5

87. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 9- 87 Quick Quiz Sections 9.4 and 9.5

88. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 9- 88 Quick Quiz Sections 9.4 and 9.5

89. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 9- 89 Quick Quiz Sections 9.4 and 9.5

90. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 9- 90 Quick Quiz Sections 9.4 and 9.5

91. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 9- 91 Quick Quiz Sections 9.4 and 9.5

92. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 9- 92 Chapter Test

93. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 9- 93 Chapter Test

94. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 9- 94 Chapter Test

95. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 9- 95 Chapter Test

96. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 9- 96 Chapter Test Solutions

97. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 9- 97 Chapter Test Solutions

98. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 9- 98 Chapter Test Solutions

99. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 9- 99 Chapter Test Solutions