Lay Analysis 5e Solutions - Chapter 8

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1、Section8.1ñConvergenceofInfiniteSeries82ThisworkisprotectedbyUnitedStatescopyrightlawsandisprovidedsolelyfortheuseofinstructorsinteachingtheircoursesandassessingstudentlearning.Disseminationorsaleofanypartofthiswork(includingontheWorldWideWeb)willdestroytheintegrityoftheworkandisnotp

2、ermitted.Theworkandmaterialsfromitshouldneverbemadeavailabletostudentsexceptbyinstructorsusingtheaccompanyingtextintheirclasses.Allrecipientsofthisworkareexpectedtoabidebytheserestrictionsandtohonortheintendedpedagogicalpurposesandtheneedsofotherinstructorswhorelyonthesematerials.Ana

3、lysiswithanIntroductiontoProof5thEditionbyStevenR.Layslay@leeuniversity.eduChapter8–InfiniteSeriesSolutionstoExercisesThefollowingnotationsareusedthroughoutthisdocument:8=thesetofnaturalnumbers{1,2,3,4,…};=thesetofrationalnumbers<
=thesetofrealnumbers"=“forevery”$=“thereexists”§=“suc

4、hthat”Section8.1–ConvergenceofInfiniteSeries1.(a)True:commentpriortoExample8.1.1.(b)True:commentpriortoExample8.1.1.2.(a)False:theharmonicseriesS1/n.(b)True:Example8.1.7.Copyright©2014PearsonEducation,Inc.Section8.1ñConvergenceofInfiniteSeries833.(a)Hintinbook:Thismayseemtrivial,butt

5、hereisreallysomethingtoprove.Recallthatthevalueofуnm=anisnotcomputedasa“sum”butasalimitofasequence.Youcannotjustadd(a1+a2+…+am–1)tobothsidesoftheequationandchangethelowerlimitoftheleftsummation.Youhavetolookatthesequenceofpartialsums.Thepointoftheexerciseistojustifyfutureuseofwhatap

6、pearstobejustasimplealgebraicmanipulation.nnÑÑSolution:Letsnk=ƒk=1aandtank=ƒkm=sothatƒn=1asnn=limandƒnm=atnn=lim.Sincetn=sn-(a1+a2+…+am-1)forním,wehaveуnm=an=limtn=(limsn)-(a1+a2+∂∂∂+am-1)Ñ=(ƒn=1aaanm)-+++()12?a-1.(b)Theproofissimilartopart(a).4.Ineachcasetheseriesisdivergentbecause

7、thetermsdon’tconvergetozero.5.Answers/hintsinbook:(a)1/2;(c)4/3;(e)1;(g)1/3;(i)1/2(k)1/4.In(k),lookatthepartialsumsandnotethat2/[n(n+1)(n+2)]=1/[n(n+1)]–1/[n+1)(n+2)].In(l),lookatthepartialsumsandtrytobreakupthetermsasin(i).Additionalanswers:(b)1/4;(d)-3/7;(f)1/2;(h)3/2;(j)1/5;(l)1/[

8、k(k!)]Deriva