ODD08 - CHAPTER Infinite Series Section 8.1 Section 8.2...

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C H A P T E R 8Infinite SeriesSection 8.1 Sequences . . . . . . . . . . . . . . . . . . . . . 121Section 8.2 Series and Convergence . . . . . . . . . . . . . . 126Section 8.3 The Integral Test and p-Series. . . . . . . . . . 131Section 8.4 Comparisons of Series. . . . . . . . . . . . . . 135Section 8.5 Alternating Series . . . . . . . . . . . . . . . . . 138Section 8.6 The Ratio and Root Tests . . . . . . . . . . . . . 142Section 8.7 Taylor Polynomials and Approximations . . . . . 147Section 8.8 Power Series . . . . . . . . . . . . . . . . . . . . 152Section 8.9 Representation of Functions by Power Series . . 157Section 8.10 Taylor and Maclaurin Series. . . . . . . . . . . 160Review Exercises. . . . . . . . . . . . . . . . . . . . . . . . .167Problem Solving. . . . . . . . . . . . . . . . . . . . . . . . .172
121C H A P T E R 8Infinite SeriesSection 8.1SequencesSolutions to Odd-Numbered Exercises1.a52532a42416a3238a2224a1212an2n3.a5125132a4124116a312318a212214a112112an12n5.a5sin 521a4sin 20a3sin 321a2sin 0a1sin 21ansin n27.a511552125a411042116a3163219a2132214a111121an1n n12n29.a551512512125a45141167716a351319439a251214194a15115an51n1n211.a5355!243120a4344!8124a3333!276a2322!92a131!3an3nn!13.2 10118a52a412 6110a42a312 416a32a212 314a22a11a13, ak12ak115.a512a41242a412a31284a312a212168a212a1123216a132, ak112ak
17.Because and the sequence matches graph (d).a282183,a1811419.This sequence decreases and Matches (c).a24 0.52.a14,21.an23n, n1, . . . , 101112823.an160.5n1, n1, . . . , 10121101825.an2nn1, n1, 2, . . . , 101211327.Add 3 to preceeding term.a63 6117a53 5114an3n129.Multiply the preceeding term by 12.a6325332an324316an32n131.9109010!8!8! 9108!33.n1n1 !n!n!n1n!35.12n2n12n1 !2n1 !2n1 !2n1 ! 2n2n137.limn5n2n22539.212limn2nn21limn211n241.limnsin1n043.The graph seems to indicate that the sequence convergesto 1. Analytically,limnanlimnn1nlimxx1xlimx11.1112345.The graph seems to indicate that the sequence diverges.Analytically, the sequence is Hence,does not exist.limnanan0, 1, 0, 1, 0, 1, . . . .1212247.does not exist (oscillates between and 1), diverges.1limn1nnn149.convergeslimn3n2n42n2132,51.convergeslimn11nn0,53.converges(L’Hôpital’s Rule)limn321n0,limnlnn32nlimn32lnnn122Chapter 8Infinite Series
55.convergeslimn34n0,57.divergeslimnn1 !n!limnn1,59.convergeslimn12nn2n0,limnn1nnn1limnn12n2n n161.convergesp> 0, n2limnnpen0,63.where convergesukn,limn1knnlimu01u1u kekan1knn65.convergeslimnsin nnlimnsin n1n0,67.an3n269.ann2271.ann1n273.an1n12n275.an11nn1n77.annn1n279.an1n1135 . . .