Calculus 2nd Edition Rogawski

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578 C H A P T E R 11 INFINITE SERIES y y 1 S 0.5 S 1S3 12 S5 10 8 6 S4 S 6 4 S 2 2 n 5 10 15 20 10 20 30 40 50 n ∞∑ (A) Partial sums of S = (−1) n−1 √ 1 n n=1 FIGURE 3 (B) Partial sums of ∞∑ n=1 1 √ n The inequality S 2N
SECTION 11.4 Absolute and Conditional Convergence 579 CONCEPTUAL INSIGHT The convergence of an infinite series ∑ a n depends on two factors: (1) how quickly a n tends to zero, and (2) how much cancellation takes place among the terms. Consider Harmonic series (diverges): 1 + 1 2 + 1 3 + 1 4 + 1 5 +··· p-Series with p = 2 (converges): 1 + 1 2 2 + 1 3 2 + 1 4 2 + 1 5 2 +··· Alternating harmonic series (converges): 1 − 1 2 + 1 3 − 1 4 + 1 5 − ··· The harmonic series diverges because reciprocals 1/n do not tend to zero quickly enough. By contrast, the reciprocal squares 1/n 2 tend to zero quickly enough for the p-series with p = 2 to converge. The alternating harmonic series converges, but only due to the cancellation among the terms. 11.4 SUMMARY • ∑ an converges absolutely if the positive series ∑ |a n | converges. • Absolute convergence implies convergence: If ∑ |a n | converges, then ∑ a n also converges. • ∑ an converges conditionally if ∑ a n converges but ∑ |a n | diverges. • Leibniz Test: If {a n } is positive and decreasing and lim n→∞ a n = 0, then the alternating series S = ∞∑ (−1) n−1 a n = a 1 − a 2 + a 3 − a 4 + a 5 − ··· n=1 converges. Furthermore, |S − S N |
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Publisher: Ruth Baruth Senior Acqui
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To Julie
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CONTENTS CALCULUS Chapter 1 PRECALC
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ABOUT JON ROGAWSKI As a successful
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x PREFACE Placement of Taylor Polyn
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xii PREFACE MEDIA Online Homework O
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xiv PREFACE FEATURES Conceptual Ins
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xvi PREFACE ACKNOWLEDGMENTS Jon Rog
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xviii PREFACE Seminole Community Co
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xx PREFACE University; Legunchim L.
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xxii PREFACE It is a pleasant task
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2 CHAPTER 1 PRECALCULUS REVIEW |a|
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4 CHAPTER 1 PRECALCULUS REVIEW y y
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6 CHAPTER 1 PRECALCULUS REVIEW Antl
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8 CHAPTER 1 PRECALCULUS REVIEW Two
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10 CHAPTER 1 PRECALCULUS REVIEW •
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12 CHAPTER 1 PRECALCULUS REVIEW 61.
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14 CHAPTER 1 PRECALCULUS REVIEW The
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16 CHAPTER 1 PRECALCULUS REVIEW Sol
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24 CHAPTER 1 PRECALCULUS REVIEW Exe
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26 CHAPTER 1 PRECALCULUS REVIEW Rad
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28 CHAPTER 1 PRECALCULUS REVIEW FIG
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30 CHAPTER 1 PRECALCULUS REVIEW c
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34 CHAPTER 1 PRECALCULUS REVIEW FIG
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36 CHAPTER 1 PRECALCULUS REVIEW fro
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38 CHAPTER 1 PRECALCULUS REVIEW Exe
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2 LIMITS Calculus is usually divide
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42 CHAPTER 2 LIMITS We cannot defin
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44 CHAPTER 2 LIMITS We conclude tha
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46 CHAPTER 2 LIMITS 3. Let v = 20
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48 CHAPTER 2 LIMITS Percent infecte
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50 CHAPTER 2 LIMITS The limit conce
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52 CHAPTER 2 LIMITS y TABLE 3 16 x
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54 CHAPTER 2 LIMITS In the next exa
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56 CHAPTER 2 LIMITS y f (y) y f (y)
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58 CHAPTER 2 LIMITS b x − 1 66. S
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60 CHAPTER 2 LIMITS (b) Use the Pro
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62 CHAPTER 2 LIMITS a x − 1 40. A
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64 CHAPTER 2 LIMITS 5 4 3 2 1 y x 1
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66 CHAPTER 2 LIMITS y y y y 2 y = x
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68 CHAPTER 2 LIMITS Temperature (°
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70 CHAPTER 2 LIMITS 50. Sawtooth Fu
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72 CHAPTER 2 LIMITS Other indetermi
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74 CHAPTER 2 LIMITS y 6 4 y = 1 x
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76 CHAPTER 2 LIMITS x − 4 35. Use
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78 CHAPTER 2 LIMITS y y y C B = (co
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80 CHAPTER 2 LIMITS 3. What does th
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82 CHAPTER 2 LIMITS y y = 2 x y y =
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84 CHAPTER 2 LIMITS 3x 3 − 7x + 9
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86 CHAPTER 2 LIMITS Exercises 1. Wh
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88 CHAPTER 2 LIMITS Altitude (m) 10
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90 CHAPTER 2 LIMITS (a) f (c) = 3 h
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92 CHAPTER 2 LIMITS 1 y y 1 0.8 0.5
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94 CHAPTER 2 LIMITS Because we are
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96 CHAPTER 2 LIMITS This proves tha
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98 CHAPTER 2 LIMITS Further Insight
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100 CHAPTER 2 LIMITS 67. In the not
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102 CHAPTER 3 DIFFERENTIATION y y y
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104 CHAPTER 3 DIFFERENTIATION Step
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106 CHAPTER 3 DIFFERENTIATION 3.1 S
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108 CHAPTER 3 DIFFERENTIATION y 5 4
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110 CHAPTER 3 DIFFERENTIATION y y 7
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112 CHAPTER 3 DIFFERENTIATION THEOR
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114 CHAPTER 3 DIFFERENTIATION EXAMP
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116 CHAPTER 3 DIFFERENTIATION GRAPH
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118 CHAPTER 3 DIFFERENTIATION 4. Ch
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120 CHAPTER 3 DIFFERENTIATION 62. S
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122 CHAPTER 3 DIFFERENTIATION REMIN
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124 CHAPTER 3 DIFFERENTIATION REMIN
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126 CHAPTER 3 DIFFERENTIATION 3.3 E
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128 CHAPTER 3 DIFFERENTIATION Furth
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130 CHAPTER 3 DIFFERENTIATION (b) T
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132 CHAPTER 3 DIFFERENTIATION • S
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134 CHAPTER 3 DIFFERENTIATION FIGUR
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136 CHAPTER 3 DIFFERENTIATION 20. T
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138 CHAPTER 3 DIFFERENTIATION F(r)
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142 CHAPTER 3 DIFFERENTIATION Exerc
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144 CHAPTER 3 DIFFERENTIATION CAUTI
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146 CHAPTER 3 DIFFERENTIATION π
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148 CHAPTER 3 DIFFERENTIATION (c) B
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150 CHAPTER 3 DIFFERENTIATION Chris
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156 CHAPTER 3 DIFFERENTIATION 4 3 2
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158 CHAPTER 3 DIFFERENTIATION Notic
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160 CHAPTER 3 DIFFERENTIATION We co
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162 CHAPTER 3 DIFFERENTIATION y y 2
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164 CHAPTER 3 DIFFERENTIATION Both
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166 CHAPTER 3 DIFFERENTIATION Step
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170 CHAPTER 3 DIFFERENTIATION Exerc
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172 CHAPTER 3 DIFFERENTIATION y (I)
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174 CHAPTER 3 DIFFERENTIATION 85. A
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176 CHAPTER 4 APPLICATIONS OF THE D
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5 THE INTEGRAL The basic problem in
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246 CHAPTER 5 THE INTEGRAL f (a + 2
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248 CHAPTER 5 THE INTEGRAL Linearit
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250 CHAPTER 5 THE INTEGRAL Computin
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252 CHAPTER 5 THE INTEGRAL 30 20 10
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254 CHAPTER 5 THE INTEGRAL • If f
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256 CHAPTER 5 THE INTEGRAL In Exerc
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258 CHAPTER 5 THE INTEGRAL c 1 c 2
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260 CHAPTER 5 THE INTEGRAL y f(c i
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262 CHAPTER 5 THE INTEGRAL y y = x
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264 CHAPTER 5 THE INTEGRAL • The
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266 CHAPTER 5 THE INTEGRAL 31. ∫
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268 CHAPTER 5 THE INTEGRAL Proof Th
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270 CHAPTER 5 THE INTEGRAL We know
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272 CHAPTER 5 THE INTEGRAL ∫ b
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274 CHAPTER 5 THE INTEGRAL Proof Fi
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276 CHAPTER 5 THE INTEGRAL y y = f(
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278 CHAPTER 5 THE INTEGRAL 2 1 0
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280 CHAPTER 5 THE INTEGRAL EXAMPLE
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282 CHAPTER 5 THE INTEGRAL In Secti
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284 CHAPTER 5 THE INTEGRAL 20. To m
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286 CHAPTER 5 THE INTEGRAL In gener
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288 CHAPTER 5 THE INTEGRAL ∫ EXAM
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290 CHAPTER 5 THE INTEGRAL 5.6 SUMM
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292 CHAPTER 5 THE INTEGRAL ∫ π/2
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294 CHAPTER 5 THE INTEGRAL ∫ 5 35
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6 APPLICATIONS OF THE INTEGRAL In t
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298 CHAPTER 6 APPLICATIONS OF THE I
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340 CHAPTER 7 EXPONENTIAL FUNCTIONS
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414 CHAPTER 8 TECHNIQUES OF INTEGRA
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9 FURTHER APPLICATIONS OF THE INTEG
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480 CHAPTER 9 FURTHER APPLICATIONS
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514 C H A P T E R 10 INTRODUCTION T
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628 C H A P T E R 12 PARAMETRIC EQU
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664 C H A P T E R 13 VECTOR GEOMETR
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( x 5 ) 2 + ( y 7 ) 2 + ( z 9 712 C
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( x a 714 C H A P T E R 13 VECTOR G
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730 C H A P T E R 14 CALCULUS OF VE
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15 DIFFERENTIATION IN SEVERAL VARIA
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782 C H A P T E R 15 DIFFERENTIATIO
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16 MULTIPLE INTEGRATION Integrals o
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868 C H A P T E R 16 MULTIPLE INTEG
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946 C H A P T E R 17 LINE AND SURFA
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1000 C H A P T E R 17 LINE AND SURF
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1010 C H A P T E R 18 FUNDAMENTAL T
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A2 APPENDIX A THE LANGUAGE OF MATHE
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B PROPERTIES OF REAL NUMBERS “The
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A10 APPENDIX B PROPERTIES OF REAL N
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A12 APPENDIX B PROPERTIES OF REAL N
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A14 APPENDIX C INDUCTION AND THE BI
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A16 APPENDIX C INDUCTION AND THE BI
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D ADDITIONAL PROOFS In this appendi
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A20 APPENDIX D ADDITIONAL PROOFS TH
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A22 APPENDIX D ADDITIONAL PROOFS Th
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A28 ANSWERS TO ODD-NUMBERED EXERCIS
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A120 REFERENCES Chapter 3 Review (E
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A122 REFERENCES Section 15.8 (EX 42
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A124 PHOTO CREDITS PAGE 410 left Li
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I2 INDEX asymptotic behavior, 81, 2
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I4 INDEX curvilinear coordinates, 9
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I6 INDEX Fourier Series, 422 fracti
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I14 INDEX temperature: directional
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Calculus 2nd Edition Rogawski

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