《高数双语》课件section 7-3

Section6.3SeriesofFunctionsandPowerSeries1Overview2ConceptsoffunctionalseriesPowerseriesTestsforconvergenceofpowerseriesOperationsofpowerseriesFunctionalSeries3Definition

(FunctionalSeries[函数项级数])Aseriesoftheform,whereisafunctionofxinasetA⊆R,iscalledaseriesoffunctions(orfunctionalseries)definedinA.Definition(ConvergenceDomain)

iftheserieswithconstanttermsconverges,thenwesaythattheseriesoffunctionsisconvergentat,andiscalledaconvergencepointoftheseriesoffunctions;ifdiverges,iscalledadivergencepoint.ThesetconsistingofalltheConvergencepointsiscalledtheConvergencedomain[收敛域]oftheseries.FunctionalSeries4Definition(SumfunctionandPointwiseConvergence)If

Distheconvergencedomainoftheseries,thentheserieshasasum.HencethesumoftheseriesisactuallyafunctionofxdefinedinD,calledthesumfunction[和函数]anddenotedbyInthiscase,theseriesissaidtoexhibiteverywhereconvergenceorpointwiseconvergence

toinD.Generally,itiscalledsimplyconvergencetoinD.FunctionalSeries5Justastheserieswithconstantterms,wecandefinethepartialsumofseries.

Definition(PartialsumandRemainder)

Thequantityiscalledthepartialsum[部分和]

oftheseries,while

iscalledaremainder[余项]

oftheseries.Itiseasytoseethatthenecessaryandsufficientcondition

foraseriestobeeverywhereconvergentinisthatorApplyingtheDefinition6Example

FindtheconvergencedomainoftheseriesSolutionSinceisaconvergenceseries,theconvergencedomainoftheseriesisFinish.TheGeometricSeries7ThesumfunctionofthegeometricseriesisTheconvergencedomainofthegivenseriesistheinterval,centeredat.Solution

Example

Discusstheconvergenceofthegeometricserieswhereisaconstant.Finditsconvergencedomainandsumfunction.Finish.ApplyingtheDefinition8Example

FindtheconvergencedomainandsumfunctionoftheseriesSolutionSincewehavethatandthelimitofdoesnotexistforHence,theconvergencedomainofthegivenseriesisfunctionisthesumFinish.PowerSeries9Definition(PowerSeries[幂级数]

)

Anexpressionoftheformisapowerseriescenteredat

.Anexpressionoftheformisapowerseriescenteredat

.Thetermisthenthterm;thenumberaisthecenter.Aserieswhosegeneraltermisapowerfunctioniscalledapowerseries.PowerSeries10Note:Thisistypicalbehavior,aswesoonsee.Apowerseriesconvergesforallx,convergesonafiniteintervalwiththesamecenterastheseries,orconvergesonlyatthecenteritself.Thegeometricseriesisapowerseriescenteredat.Thegeometricseriesconvergestoontheinterval,alsocenteredat.PowerSeries11ExampleThepowerseries(1)iscenteredatwithcoefficientsTheseriesconvergesforor.ThesumissoThisisageometricserieswithfirstterm1andratio.PowerSeries12Series(1)generatesusefulpolynomialapproximationsofforvaluesofxnear2.(1)ConvergenceofPowerSeries13thesequenceisbounded,thatis,suchthatProofofPart(1)Sinceconverges,.Thusforall

Abel’sTheorem

Considerthepowerseries.

(1)Ifitconvergesat,,thenitmustconvergeabsolutelyintheinterval;

(2)Ifitdivergesat,,thenitmustdivergein.ConvergenceofPowerSeries14ProofofPart(1)(continued)Thus,whilethegeometricseriesconvergesprovidedbecauseAccordingtothecomparisontest,thepowerseriesconvergesforPart(2)canbeprovedbycontradiction.Wedonotmentionhere.TheRadiusandIntervalofConvergence15Theorem(TheConvergenceTheoremforPowerSeries)1.ThereisapositiveRsuchthattheseriesdivergesforbutconvergesfor.Theseriesmayormaynotconvergeateitheroftheendpointsand.2.Theseriesconvergesforevery.3.Theseriesconvergesatanddivergeselsewhere

.Therearethreepossibilitiesforwithrespecttoconvergence.TheRadiusandIntervalofConvergence16ThenumberRistheradiusofconvergence[收敛半径],theopeninterval(a-R,a+R)iscalledtheconvergenceinterval[收敛区间],andthesetofallvaluesofxforwhichtheseriesconvergesistheconvergencedomain[收敛域].TheradiusofconvergencecompletelydeterminesthedomainofconvergenceifRiseitherzeroorinfinite.For,however,thereremainsthequestionofwhathappensattheendpointsoftheinterval.Thenextexampleillustrateshowtofindthedomainofconvergence.DefinitionNote:FindingtheDomainofConvergence17Example

Forwhatvaluesofxdothefollowingpowerseriesconverge?(d)(a)(b)(c)FindingtheDomainofConvergence18At,wegetthealternatingharmonicseries,At,weget,thenegativeoftheharmonicSolutionApplytheRatioTesttotheseries,whereisthenthtermoftheseriesinquestion.(a)ItdivergesifTheseriesconvergesabsolutelyfor.becausethenthtermdoesnotconvergetozero.series;itdiverges.Series(a)convergesforanddivergeselsewhere.whichconverges.FindingtheDomainofConvergence19Solution(continued)ItdivergesifTheseriesconvergesabsolutelyfor.becausethenthtermdoesnotconvergetozero.At,wegetAt,weget,itisagainconvergent.theseries,whichconvergesbytheSeries(b)convergesforanddivergeselsewhere.(b)Leibniz’sTheorem.FindingtheDomainofConvergence20Solution(continued)(c)foreveryThisseriesconvergesabsolutelyforallx.FindingtheDomainofConvergence21Solution(continued)foreveryThisseriesdivergesforallvaluesofxexceptx=0.(d)StepsforFindingtheDomainofConvergence22Step1:UsetheRatioTest(ornth-RootTest)tofindtheintervalwheretheseriesconvergesabsolutely.Ordinarily,theisanopenintervalorStep2:Iftheintervalofabsoluteconvergenceisfinite,testforconvergenceordivergenceat

eachendpoint,asinpreviousexample(a)and(b).UseaComparisonTest,theIntegralTest,ortheAlternatingSeriesTest.Step3:Iftheintervalofabsoluteconvergenceistheseriesdivergesfor(itdoesnotevenconvergeconditionally),becausethenthtermdoesnotapproachzeroforthosevalueofx.FindingtheDomainofConvergence23Theorem(Page318)

IfwhereisthecoefficientsofthepowerseriesthentheradiusofconvergenceisgivenbyFindingtheDomainofConvergence24Example

Findtheradiusofconvergence,convergenceintervalandconvergencedomainofthefollowingpowerseries:(a)(b)(c)FindingtheDomainofConvergence25Solution(a)LetandtheconvergenceSo,theradiusofconvergenceoftheseries(a)isR=2Atx=2,theseriesbecomesintervaloftheseries(a)iswhichconvergesbytheLeibniz’sHence,theconvergencedomainofthegivenseries(a)isTheorem.Atx=-2,theseriesbecomesitisdivergent.FindingtheDomainofConvergence26Solution(continued)(b)LetwehavethatSo,theradiusofconvergenceoftheseries(b)isRandtheconvergenceintervaloftheseries(a)isFindingtheDomainofConvergence27AttheseriesbecomesSolution(continued)(b)TheseriesAttheseriesbecomesThus,theseries(b)convergesatandtheseriesareconvergent.itisdivergent.Hence,theconvergencedomainofthegivenseries(b)isFindingtheDomainofConvergence28Solution(continued)(c)Theseriescontainsonlyevenorderpowers.LetTheseries(c)becomesLetTheseriesconvergesabsolutelyforanddivergesforTherefore,theseries(c)convergesabsolutelyforanddivergesforandtheconvergenceradiusofconvergenceoftheseries(a)isRintervaloftheseries(c)isTheAttheseriesbecomeitisconvergent.Hence,theconvergencedomainofthegivenseries(c)isOperationsofPowerSeries29Theorem(AlgebraicOperation)Ifandconvergeabsolutelyforand,respectively,and.(1)Thelinearcombinationconvergesinthecommonconvergenceintervaland(2)TheproductseriesalsoconvergesinandwhereOperationsofPowerSeries30Theradiusofconsequenceofthesumandproductoftwopowerseriesmaybelargerthantheradiiand,where.Note:Itiseasytoseethattheseriesandarebothconvergewith,buttheradiioftheirsumis.Inthiscase,ofcourse,theAdditionTheoremholds.ExampleApplyingtheMultiplicationTheorem31Example

Multiplythegeometricseriesforbyitselftogetapowerseriesfor,for.SolutionLetandApplyingtheMultiplicationTheorem32Then,bytheSeriesMultiplicationTheorem,istheseriesof.TheseriesallconvergeabsolutelyforSolution(continued)Finish.Example

Multiplythegeometricseriesforbyitselftogetapowerseriesfor,for.OperationsofPowerSeries33Theorem

Supposeconvergesfor|x|<R(R>0)anditssumfunctionisS(x).Then(1)S