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1、Section4.1ñConvergence43ThisworkisprotectedbyUnitedStatescopyrightlawsandisprovidedsolelyfortheuseofinstructorsinteachingtheircoursesandassessingstudentlearning.Disseminationorsaleofanypartofthiswork(includingontheWorldWideWeb)willdestroytheintegrityoftheworkandisnotpermitted.Theworkandmateri
2、alsfromitshouldneverbemadeavailabletostudentsexceptbyinstructorsusingtheaccompanyingtextintheirclasses.Allrecipientsofthisworkareexpectedtoabidebytheserestrictionsandtohonortheintendedpedagogicalpurposesandtheneedsofotherinstructorswhorelyonthesematerials.AnalysiswithanIntroductiontoProof5thE
3、ditionbyStevenR.Layslay@leeuniversity.eduChapter4–SequencesSolutionstoExercisesThefollowingnotationsareusedthroughoutthisdocument:8=thesetofnaturalnumbers{1,2,3,4,…};=thesetofrationalnumbers<=thesetofrealnumbers"=“forevery”$=“thereexists”§=“suchthat”Section4.1-Convergence1.(a)False:SeeExampl
4、e16.1(a).(b)True:Definition4.1.2.n(c)False:sn=1/nandtn=1/2.(d)True:Theorem4.1.13.2.(a)True:Definition4.1.2andtheinequalitysnÇ
5、sn
6、.Copyright©2014PearsonEducation,Inc.Section4.1ñConvergence44(b)False:Letsn=-1foralln.(c)False:Theorem4.1.8requiresthatanç0.(d)True:Theorem4.1.14.3.Answersinbook:(a)
7、1,4,9,16,25,36,49;(c)11,,1---,,,1111,.22222Additionalanswers:(b)----1,,11,11,,,11;(d)11,,,,,,3791113.2345674721316192233324.Letk=7andm=5.Tomake6n+17nÇ7n,wewantní17n,ní17,andní5.Notethatanyvaluegreaterthan6willworkfork,anddifferentvaluesforkwillyielddifferentvaluesform.33325.Letk=1/2andm=4.Tom
8、aken–7ní(1/2)n,wewant(1/2)ní7n,ní14,andní4.Notethatanypositivefractionlessthan1willworkfork,anddifferentvaluesforkwillyielddifferentvaluesform.6.(a)Givene>0,takeN>
9、k
10、/e.1/kkk(b)Givene>0,takeN>(1/e).Thenn≥Nimpliesn>1/e,so(1/n)0,takeN>11/e.nn++33nsinn1(d)-Ç0,sogiven
11、e>0,takeN>1/e.nnnn+324221122(e)-Ç0=whenní5,sincenn-í13iffní13iffní26iffní6.nn221n22-132Sogivene>0,takeN>max{4/e,6}.nn++222n4221122(f)-<0Ç=whenn≥3,sincennn-í3iffí3iffní6iffní3.Sogivennn22n1n2n22+-33-2e>0,takeN>max{4/e,3}.11⊬·⊬·17.(a)Ç∆¸∆¸ç0byT
