数学分析 高等数学 微积分 英语课件 上海交通大学 chapter11b

ThecomparisontestsTheoremSupposethatandareserieswithpositiveterms,then(i)Ifisconvergentandforalln,thenisalsoconvergent.(ii)Ifisdivergentandforalln,thenisalsodivergent.Ex.Determinewhetherconverges.Sol.Sotheseriesconverges.ThelimitcomparisontestTheoremSupposethatandareserieswithpositiveterms.SupposeThen(i)whencisafinitenumberandc>0,theneitherbothseriesconvergeorbothdiverge.(ii)whenc=0,thentheconvergenceofimpliestheconvergenceof(iii)whenthenthedivergenceofimpliesthedivergenceofExampleEx.Determinewhetherthefollowingseriesconverges.Sol.(1)diverge.choosethen(2)diverge.takethen(3)convergeforp>1anddivergefortakethenQuestionEx.Determinewhethertheseriesconvergesordiverges.Sol.AlternatingseriesAnalternatingseriesisaserieswhosetermsarealternativelypositiveandnegative.Forexample,Then-thtermofanalternatingseriesisoftheformwhereisapositivenumber.ThealternatingseriestestTheoremIfthealternatingseriessatisfies(i)foralln(ii)Thenthealternatingseriesisconvergent.Ex.Thealternatingharmonicseries

isconvergent.ExampleEx.Determinewhetherthefollowingseriesconverges.Sol.(1)converge(2)convergeQuestion.AbsoluteconvergenceA

seriesiscalledabsolutelyconvergentiftheseriesofabsolutevaluesisconvergent.Forexample,theseriesisabsolutelyconvergentwhilethealternatingharmonicseriesisnot.A

seriesiscalledconditionallyconvergentifitisconvergentbutnotabsolutelyconvergent.Theorem.Ifaseriesisabsolutelyconvergent,thenitisconvergent.ExampleEx.Determinewhetherthefollowingseriesisconvergent.Sol.(1)absolutelyconvergent(2)conditionallyconvergentTheratiotestTheratiotest(1)If

thenisabsolutelyconvergent.(2)Iforthendiverges.(3)Iftheratiotestisinconclusive:thatis,noconclusioncanbedrawnabouttheconvergenceofExampleEx.TesttheconvergenceoftheseriesSol.(1)convergent(2)convergentfordivergentforTheroottestTheroottest(1)If

thenisabsolutelyconvergent.(2)Iforthendiverges.(3)Iftheroottestisinconclusive.ExampleEx.TesttheconvergenceoftheseriesSol.convergentfordivergentforRearrangementsIfwerearrangetheorderoftheterminafinitesum,thenofcoursethevalueofthesumremainsunchanged.Butthisisnotthecaseforaninfiniteseries.Byarearrangementofaninfiniteserieswemeanaseriesobtainedbysimplychangingtheorderoftheterms.Itturnsoutthat:ifisanabsolutelyconvergentserieswithsum,thenanyrearrangementofhasthesamesum.However,anyconditionallyconvergentseriescanberearrangedtogiveadifferentsum.ExampleEx.ConsiderthealternatingharmonicseriesMultiplyingthisseriesbywegetorAddingthesetwoseries,weobtainStrategyfortestingseriesIfwecanseeataglancethatthendivergenceIfaseriesissimilartoap-series,suchasanalgebraicform,oraformcontainingfactorial,thenusecomparisontest.Foranalternatingseries,usealternatingseriestest.StrategyfortestingseriesIfn-thpowersappearintheseries,useroottest.Iffdecreasingandpositive,useintegraltest.Sol.(1)diverge(2)converge(3)diverge(4)convergePowerseriesApowerseriesisaseriesoftheformwherexisavariableandareconstantscalledcoefficientsofseries.Foreachfixedx,thepowerseriesisausualseries.Wecantestforconvergenceordivergence.Apowerseriesmayconvergeforsomevaluesofxanddivergeforothervaluesofx.SothesumoftheseriesisafunctionPowerseriesForexample,thepowerseriesconvergestowhenMoregenerally,Aseriesoftheformiscalledapowerseriesin(x-a)orapowerseriescenteredataorapowerseriesabouta.ExampleEx.Forwhatvaluesofxisthep