课程介绍

 

　　CalculusTwo:SequencesandSeriesisanintroductiontosequences,infiniteseries,convergencetests,andTaylorseries.Thecourseemphasizesnotjustgettinganswers,butaskingthequestion"whyisthistrue?"

 

　　课程目录

 

　　1Sequences

 

　　1.1HowCanISucceedinThisCourse

 

　　1.2.1WhatisaSequence

 

　　1.2.2HowisaSequencePresented

 

　　1.2.3CantheSameSequencebePresentedinDifferentWays

 

　　1.3.1HowCanWeBuildNewSequencesfromOldSequences

 

　　1.3.2WhatisanArithmeticProgression

 

　　1.4.1WhatisanGeometricProgression

 

　　1.4.2WhatistheLimitofaSequence

 

　　1.4.3Visually,WhatistheLimitofaSequence

 

　　1.5.1IsitEasytoFindtheLimitofaSequence

 

　　1.5.2ForSomeEpsilon,HowLargeNeedNBe

 

　　1.6.1HowDoSequencesHelpwiththeSquareRootofTwo

 

　　1.6.2WhenisaSequenceBounded

 

　　1.6.3WhenisaSequenceIncreasing

 

　　1.6.4WhatistheMonotoneConvergenceTheorem

 

　　1.6.5HowCantheMonotoneConvergenceTheoremHelp

 

　　1.7.1IsThereaSequenceThatIncludesEveryInteger

 

　　1.7.2IsThereaSequenceThatIncludesEveryRealNumber

 

　　2Overview

 

　　2.1WhatHappensinThisModule

 

　　2.2WhatDoes∑a?=LMean_

 

　　2.3WhatisaGeometricSeries_

 

　　2.4WhatistheValueof∑???∞r?_

 

　　2.5WhyDoes∑???∞1_2?=2_

 

　　2.6WhatistheSumofaTelescopingSeries_

 

　　2.7DoestheSeries∑n_(n+1)ConvergeorDiverge_

 

　　2.8DoestheSeries1+1_2+1_3+?ConvergeorDiverge_

 

　　2.9Does∑sin?k_2?ConvergeorDiverge_

 

　　2.10WhatistheComparisonTest_

 

　　2.11HowCanGroupingMaketheComparisonTestEvenBetter_

 

　　2.12Whatis∑1_n?_

 

　　2.13InWhatSenseDoes0.99999?Equal1_

 

　　2.14InWhatSenseis∑9?10?Meaningful_

 

　　3Overview

 

　　3.1WhatWillHappeninThisModule_

 

　　3.2DoesSumn^5_4^nConverge_

 

　　3.3WhatDoestheRatioTestSay_

 

　　3.4DoestheRatioTestAlwaysWork_

 

　　3.5DoesSumn!_n^nConverge_

 

　　3.6HowDoesn!Compareton^n_

 

　　3.7WhyDon'tILovetheRootTest_

 

　　3.8HowCanIntegratingHelpUstoAddressConvergence_

 

　　3.9HowElseCanIShowtheHarmonicSeriesDiverges_

 

　　3.10DoesSum1_n^pConverge_

 

　　3.11DoesSum1_(nlogn)Converge_

 

　　3.12HowFarOutCanYouBuildaOneSidedBridge_

 

　　4Overview

 

　　4.1WhatisthisModuleAllAbout_

 

　　4.2WhyHaveWeBeenAssumingtheTermsarePositive_

 

　　4.3WhyDoAbsolutelyConvergentSeriesJustPlainConverge_

 

　　4.4WhyisAbsoluteConvergenceanImportantConcept_

 

　　4.5WhatisConditionalConvergence_

 

　　4.6WhatisanAlternatingSeries_

 

　　4.7WhatistheAlternatingSeriesTest_

 

　　4.8WhyCanPeopleGetAwayWithWritingsum_na_n_

 

　　4.9WhyisThisAllsoVague...orCoarse_

 

　　4.10WhatHappensifIrearrangetheTermsinaConditionallyConvergentSeries_

 

　　5Overview

 

　　5.1WhatarePowerSeries_

 

　　5.2ForWhichValuesDoesaPowerSeriesConverge_

 

　　5.3WhyDoesaPowerSeriesConvergeAbsolutely_

 

　　5.4HowComplicatedMighttheIntervalofConvergenceBe_

 

　　5.5HowDoIFindtheRadiusofConvergence_

 

　　5.6WhatiftheRadiusofConvergenceisInfinite_

 

　　5.7WhatiftheRadiusofConvergenceisZero_

 

　　5.8WhatisaPowerSeriesCenteredArounda_

 

　　5.9CanIDifferentiateaPowerSeries_

 

　　5.10CanIIntegrateaPowerSeries_

 

　　5.11WhyMightIbelieveIHaveaPowerSeriesfore^x_

 

　　5.12WhatHappensifIMultiplyTwoPowerSeries_

 

　　5.13WhatHappensifITransform1_(1-x)_

 

　　5.14WhatisaFormulafortheFibonacciNumbers_

 

　　6Overview

 

　　6.1WhatisThisLastModuleAbout_

 

　　6.2WhatisBetterThanaLinearApproximation_

 

　　6.3WhatistheTaylorSeriesforfAroundZero_

 

　　6.4WhatistheTaylorSeriesforfCenteredArounda_

 

　　6.5WhatistheTaylorSeriesforSinAroundZero_

 

　　6.6WhatisTaylor'sTheorem_

 

　　6.7WhyistheRadiusofConvergenceof1_(1+x^2)soSmall_

 

　　6.8HowisTaylor'sTheoremLikeaSoupedUpVersionoftheMeanValueTheorem_

 

　　6.9Approximately,WhatiscosxWhenxisNearZero_

 

　　6.10HowDoTaylorSeriesProvideIntuitionForLimits_

 

　　6.11WhatisaRealAnalyticFunction_

 

　　6.12HowAreRealAnalyticFunctionsSometimeslikeHolograms_