calculus II-1

1 CalculusIISemester2 2014 2015 2 Instructor Dr YeHuajunT A MissLiuKaihuiOffice E409Tel 3620622 office 3620630 TA Email hjye uic edu hk Instructor lisaliu uic edu hk TA 3 4 ScoreSystem 5 SomenoticesonthisCourse Assignmentsmustbehandedinbeforethedeadline Afterthedeadline werefusetoacceptyourassignments Forthemid termtestandfinalexamination youcannotbringanythingexceptcalculator somestationeriesandwater Mobilearenotallowed Forthefinalexamination wecannottellyouthescorebeforetheARinformtheofficialresults Ifyouhaveanyquestiononthescore youcancheckthemarkedsheetviaAR 6 GeneralInformation TextbookCalculus EarlyTranscendentalFunctions3rdEditionSmithandMinton2007 McGrawHill InternationalEditionAdvantagestextbookfortwosemestersMoreapplications 7 GeneralInformation ReferencesS Salas E HilleandG J Etgen Calculus OneandSeveralVariables 8thedition JohnWiley Sons 1999 L D Hoffmann G L Bradley Calculus forBusiness EconomicsandtheSocialandLifesciences 9thedition McGrawHill 2004 J Stewart Calculus 4thedition Books Cole 1999 D Hughes Hallett A M Gleason W G McCallumetal Calculus SingleandMultivariable 2ndedition Wiley 1998 8 Chapter8InfiniteSeries 无穷级数 InthisChapter wewillknowsomeimportantconcepts InfiniteSeries 无穷级数 PowerSeries 幂级数 TaylorSeries 泰勒级数 FourierSeries 傅立叶级数 9 Section8 1SequencesofRealNumbers DefinitionofSequences 序列 Asequenceisalistofnumberswritteninadefiniteorder Thenotationis or or 10 Examples 11 DefinitionAsequenceconvergestoafinitenumberLifapproachesLwhenngoestoinfinity Wewriteandcallthesequenceconvergent 收敛 Wecallasequencedivergent 发散 ifitdoesnotapproachanyfinitenumberwhenngoestoinfinity 12 definitionoflimitsofsequence Givenanynumber thereexistsanintegerN suchthateveryn NimpliesQuestion definitionofdivergenceofsequence 13 CauchyTheorem 柯西收敛定理 Thesequenceshaveafinitelimitifandonlyifforany thereexistsanumberN 0 suchthatanyn N andn Nimplies 14 SUBSEQUENCES 子序列 Bychoosinginfinitelymanytermsfromasequence wegetasubsequence Thenewlyformedsequenceusuallyiswrittenas i e Theindicessatisfyand 15 Proposition IfasequenceconvergestoL theneverysubsequenceconvergestoL ExampleConsiderthesequence Thesubsequenceconvergesto 1 andthe subsequenceconvergesto1 Sincethetwosubsequenceshavedifferentlimits abovepropositiontellsusthatthesequencediverges 16 Exercise1 17 18 19 20 21 22 23 Exercise2 24 Exercise3 25 Exercise4 26 27 Exercise5 28 Section8 2InfiniteSeries 无穷级数 29 InfiniteSeries 无穷级数 ConsiderthesequenceSupposewestartaddingthetermstogether Wedefinethepartialsumsby 30 31 32 GeometricSeries 几何序列 33 34 35 Exercise6 36 Question 37 HarmonicSeries 调和序列 38 39 Exercise7 40 Section8 3TheIntegralTestandComparisonTests 积分判别法与比较判别法 Foragivenseries supposethatthereisafunctionfforwhichwherefiscontinuousanddecreasingandf x 0forallx 1 Weconsiderthenthpartialsum 41 42 IntegralTest 积分判别法 43 44 Exercise8 45 TheRemainderWedefinetheremaindertobe 46 47 ComparisonTests 比较判别法 Remark Wealwaysusesomenotableseriesforcomparison forexample geometricseries harmonicseriesandp series 48 49 Exercise9 50 LimitComparisonTest 51 52 Exercise10 53 Section8 4AlternatingSeries 交错级数 Analternatingseriesisanyseriesoftheformwhereforallk 54 AlternatingSeriesTest 交错级数判别法 55 56 EstimatingtheSumofanAlternatingSeries 57 58 Exercise11 59 Section8 5AbsoluteConvergenceandtheRatioTest 绝对收敛与比率判别法 AbsoluteandConditionalConvergence 60 61 62 TheRatioTest 63 64 Exercise12 65 TheRootTest 根式判别式 66 Exercise13b c d f g h 67 68 Section8 6PowerSeries 幂级数 69 ThePrimarytoolforinvestigatingtheconvergenceordivergenceofapowerseriesistheRatioTest 70 RadiusofConvergence 71 72 Remark 73 Exercise14 74 DifferentiationandIntegrationofpowerseries 75 76 Exercise15 77 Section8 7TaylorSeries 泰勒级数 78 TaylorSeries 泰勒级数 Supposethatwestartwithaninfinitelydifferentiablefunction f i e fcanbedifferentiatedinfinitelyoften Then wecanconstructtheseriescalledaTaylorseriesexpansionforf MaclaurinSeries 麦克劳林级数 79 Therearetwoimportantquestionsweneedtoanswer Doesaseriesconstructedinthiswayconverge Ifso whatisitsradiusofconvergence Iftheseriesconverges itconvergestoafunction Doesitconvergetof 80 81 TaylorPolynomialofdegreen n阶泰勒多项式 82 83 Taylor sTheorem 泰勒定理 84 85 86 87 SomecommonTaylorSeries 88 Section8 8FourierSeries 傅立叶级数 FourierSeries 89 Thereareanumberofimportantquestionswemustaddress WhatfunctionscanbeexpandedinaFourierse