《高数双语》课件section 7-2

Section6.2ConvergenceTestsforSerieswithConstantTerms1ConvergenceTestsforSeriesofPositiveTerms2Definition(seriesofpositiveterms)Aseriesiscalledaseriesofpositiveterms[正项级数](orseriesofnon-negativeterms)orsimplypositiveseriesifthegeneralterman≥0,n=1,2,···.Theorem(NecessaryandSufficientCondition)Aseriesofpositivetermsisconvergentiffthesequenceofitspartialsumsisboundedabove.Thepartialsumsarenon-decreasingbecauseand3IntegralTestExampleShowthattheseriesconverges.SolutionWedeterminetheconvergenceofthisseriesbycomparingitwithThinkofthetermsoftheseriesasvaluesofthefunctionandinterpretthesevalueasareaofrectanglesunderthecurve4IntegralTestSolution(continued)Thusthepartialsumoftheseriesareboundedfromabove(by2)andtheseriesconverges.ThesumoftheseriesisknowntobeItiseasytoseethat5IntegralTestTheorem(IntegralTest[积分判别法])Let{an}beasequenceofpositiveterms.Supposethatan=f(n),where

f

isacontinuous,positive,decreasingfunctionofxforallx≥N(Nisapositiveinteger).Thentheseriesandtheintegralbothconvergeorbothdiverge.6ApplyingtheIntegralTestDoesconverge?SolutionTheintegraltestappliesbecausepositive,decreasingfunctionofxforx>1.WehaveTheintegralconverges,somusttheseries.Exampleisacontinuous,7p-SeriesandHarmonicSeriesTheIntegralTestcanbeusedtosettlethequestionofconvergenceforanyseriesoftheform,parealconstant.Suchseriesarecalledp-series.Thep-series(p

arealconstant)convergesifp>1anddivergesif

p

≤1.Thep-serieswithp=1istheharmonicseries,anditisprobablythemostfamousdivergentseriesinmathematics.8DirectComparisonTestTheDirectComparisonTestSupposeandarepositiveseriesand∃N∈N+,suchthatforalln>N.(b)

divergesifdiverges.(a)

convergesifconverges.9DirectComparisonTestInpart(a),thepartialsumsofareboundedabovebyTheythereforeformanon-decreasingsequencewithalimitSo,ifisaconvergentseries,thenconverges.ProofProofSupposeandarepositiveseriesand∃N∈N+,suchthatforalln>N.(b)divergesifdiverges.(a)convergesifconverges.10DirectComparisonTestProofInpart(b),thepartialsumsofarenotboundedfromabove.Iftheywere,thepartialsumsforwouldbeboundedbyandwouldhavetoconvergeinsteadofdiverge.ProofSupposeandarepositiveseriesand∃N∈N+,suchthatforalln>N.(b)divergesifdiverges.(a)convergesifconverges.11DirectComparisonTestToapplytheDirectComparisonTesttoaseries,weneednotincludetheearlytermsoftheseries.WecanstartthetestwithanyindexNprovidedthatweincludeallthetermsoftheseriesbeingtestedfromthereon.ProofSupposeandarepositiveseriesand∃N∈N+,suchthatforalln>N.(b)divergesifdiverges.(a)convergesifconverges.12ApplyingtheDirectComparisonTestDoesthefollowingseriesconverge?SolutionWeignorethefirstfourtermsandcomparetheremainingtermswiththoseoftheconvergentgeometricseries.WeseethatTherefore,theoriginalseriesconvergesbytheDirectComparisonTest.ToapplytheDirectComparisonTest,weneedtohaveonhandalistofserieswhoseconvergenceordivergenceweknow.Thenexttableshowsalistofwhatweknowsofar:Example13SomeImportantConvergentandDivergentSeriesConvergentSeriesDivergentSeriesGeometricSerieswithTheharmonicseriesAnyp-serieswithGeometricSerieswithTelescopingSerieslikeTheSeriesAnyp-serieswithAnyseriesforwhichthedoesnotexistor14ApplyingtheDirectComparisonTestExample

Discusstheconvergenceofthefollowingseries:Since∀n∈N+,andthegeometricseries

converges,bytheDirectComparisonTest,the

seriesconvergesaswell.Solution15ApplyingtheDirectComparisonTestSolution(2)Since∀n∈N+,andtheharmonicseriesdiverges,bytheDirectComparisonTest,divergesaswell.Example

Discusstheconvergenceofthefollowingseries:16LimitComparisonTestTheLimitComparisonTestSupposeandarepositiveseries,andbn>0for∀n∈N+,

(1)If,thenthetwoseriesconvergeordivergesimultaneously;(2)Ifandconverges,thenconverges;(3)Ifλ=+∞

anddiverges,thendiverges.17LimitComparisonTestThus,forBytheDirectComparisonTest,wehavetheconclusion(1).ProofofPart1Since,thereexistsanintegerNsuchthatfor(1)If,thenthetwoseriesconvergeordivergesimultaneously;18UsingtheLimitComparisonTest(d)(a)(b)Example

Discusstheconvergenceofthefollowingseries:(c)19UsingtheLimitComparisonTest(a)Solutionlike,soweletLetFornlarge,weexpectantobehaveSincedivergesanddivergesbypart1oftheLimitComparisonTest.20UsingtheLimitComparisonTestSolutionLetlike,soweletFornlarge,weexpectantobehaveSinceconvergesandconvergesbypart1oftheLimitComparisonTest.(b)21UsingtheLimitComparisonTest(c)SolutionLetSincetobehavelike,weletweexpectanSinceconvergesandconvergesbypart1oftheLimitComparisonTest.22UsingtheLimitComparisonTestSolutionLetFornlarge,weexpectantobehaveSincedivergesanddivergesbypart3oftheLimitComparisonTest.(d)like,whichisgreaterthanfor,sowetake23RatioTestTheRatioTestmeasurestherateofgrowth(ordecline)ofaseriesbyexaminingtheratio.Forageometricseries,therateisaconstant(),andtheseriesconvergesifandonlyifItsratioislessthat1inabsolutevalue.TheRatioTestisapowerfulruleextendingthatresult.TheRatioTest(D’Alembert’sTest)Letbeaseriesofpositivetermsandsupposethat.Then

(1)theseriesconvergesif.

(2)theseriesdivergesif.

(3)thetestisinconclusiveif.24RatioTestProofofRatioTestPart(1)Letqbeanumberbetweenand1.SinceThatis,…Theseinequalitiesshowthatthetermsofourseries,afterNthterm,approachzeromorerapidlythanthetermsinageometricserieswithratioThenthenumberispositive.25RatioTestProofofRatioTestMoreprecisely,considertheseries,whereforandforallm,andThegeometricseriesconvergesbecause,thenSince,alsoconverges.(continued)Part(1)converges.26RatioTestProofofRatioTestPart(3)FromsomeindexNonN+,andThetermsoftheseriesdonotapproachzeroasnbecomesinfinite,andtheseriesdivergesbythenth-TermTest.Part(2)Thetwoseriesandshowthatsomeothertestforconvergencemustbeusedwhen.For:For:27UsingtheRatioTest(a)(b)Example

Discusstheconvergenceofthefollowingseries:Solution(1)BytheRatioTest,sincetheseriesdiverges.(2)BytheRatioTest,theseriesconverges.28RootTestThenth-RootTestisanotherusefultoolforansweringthequestionofconvergenceforserieswithnonnegativeterms.Westatetheresultherewithoutproof.Thenth-RootTest(Cauchy’sTest)Letbeaserieswithfor,andsupposethat

Then

(1)theseriesconvergesif.

(2)theseriesdivergesif.

(3)thetestisinconclusiveif.Exercise

Whichofthefollowingseriesconverge,andwhichdiverge?(a)(b)29AlternatingSeriesDefinition(AlternatingSeries)Aseriesinwhichthetermsarealternatelypositiveandnegativeisanalternatingseries[交错级数].Forexample,30AlternatingSeriesTestTheorem(Leibniz’sTheorem)

TheseriesofLeibnizformconverges,anditssumDefinition(SeriesofLeibnizForm)

ThealternatingseriesisaseriesofLeibnizForm,if(1)forall,forsomeintegerN.(2)AlternatingSeriesTest31ProofIfnisaneveninteger,say,thenthesumofthefirsttermisSinceisnondecreasingandboundedfromabove,ithasalimit,Ifnisanoddinteger,say,thenthesumofthefirstntermsis.Since,andas,(2)(1)Combiningtheresultsofequation(1)and(2)gives32AlternatingSeriesTestExample

Discusstheconvergenceoftheseries(1)(2)Solution1)Let.ThesequenceismonotonedecreasingandsothegivenseriesisofLeibnizform.Itisconvergent.2)Let.ThesequenceismonotonedecreasingandsothegivenseriesisofLeibnizform.Itisconvergent.33GeneralSeriesandTestsforConvergenceDefinition(ConditionalConvergence[条件收敛])

Aseriesconvergesconditionally

ifitconvergesbutdoesnotconvergeabsolutely.Definition(AbsoluteConvergence[绝对收敛])

Theseries

convergesabsolutely

(isabsolutelyconvergent)ifthecorrespondingseriesofabsolutevalues,,converges.Absoluteconvergenceisimportantfortworeasons.First,wehavegoodtestsforconvergenceofseriesofpositiveterms.Second,ifaseriesconvergesabsolutely,thenitconverges.34AbsoluteConvergenceTestTheorem(TheAbsoluteConvergenceTest)

Ifconverges,thenconverges.ProofForeachn,soIfconverges,thenconvergesand,bytheDirectComparisonTest,letsusexpresstheseriesasthedifferenceofthenonnegativeseriesconverges.TheequalityTherefore,converges.twoconvergentseries:35AbsoluteConvergenceTestExampleThealternatingseriesisconvergesbecauseitconvergesabsolutely.Theorem(TheAbsoluteConvergenceTest)

Ifconverges,thenconverges.AbsoluteConvergenceTest36Example

Discusstheconvergenceofthefollowingseries;ifitconverges,isitabsolutelyorconditionallyconvergent?Solution(1)Ifx=0,obviously,theseriesconverges.Ifx≠0,set.Sincetheseriesconverges.So,thegivenseriesconvergesabsolutely.AbsoluteConvergenceTest37SolutionSetand.(2)ItiseasytoseethegivenseriesisofLeibnizform.converges.Example

Discusstheconvergenceofthefollowingseries;ifitconverges,isitabsolutelyorconditionallyconvergent?AbsoluteConvergenceTest38Solutionso(2)LetTheseriesdiverges,sotheseriesisconditionallyconvergent.Example

Discusstheconvergenceofthefollowingseries;ifitconverges,isitabsolutelyorconditionallyconverg