泰勒级数的若干展开方法_曹倩倩

20167303YINSHANACADEMICJOUＲNALJul．2016Vo1．30No．3*?234000。。O173．1A1004－1869201603－0020－041。11fxf'x…fnx…2fxx=x0fx0f'x0…fnx0…3fx0+f＇x0x－x0+fx02x－x02+…+fnx0nx－x0n+…4Ｒnxx0Ux0。、23。。22．1。1exx=1。exx=0et=1+t+t22+…+tnn+…t＜∞ex=ex－1+1=e·ex－1t=x－1ex=e∑∞n=0x－1nnx－1＜∞。2．211－x=1+x+x2+…+xn+…x＜1。2fx=13x－2x=2。02*2016－01－041987－。DOI:10.13388/j.cnki.ysajs.20160408.01513x－2=13x－2+4=14·11+34x－2=14·11－－34x－2=14－342x－2+3243x－22－3344x－23+…=∑∞n=0－1n3n4n+1x－2nx－2＜43。2．3“”。311－x=∑∞n=0xnx＜111－xnn=23…x=0。11－x2=ddx11－x=ddx∑∞n=0xn=∑∞n=0ddxxn=∑∞n=1nxn－1=∑∞n=0n+1xnx＜1…11－xk+1=1k·ddx11－xk=1k∑∞n=0ddxn+k－1…n+1k－1xn=∑∞n=0n+k…n+1kxnx＜1。2．4“”。4fx=arctanxx=0。arctanx=∫0x11+x2dx11+x2=1－x2+x4－x6+…=∑∞n=0－1nx2nx＜1arctanx=∫0x11+x2dx=∫0x∑∞n=0－1nx2ndx=∑∞n=0∫0x－1nx2ndx=∑∞n=0－1n2n+1x2n+1x＜1。2．5。。5fx=exln1+xx=0。ex=1+x+x22+…+xnn+…x＜∞ln1+x=x－x22+x33+…+－1n－1xnn+…x＜1exln1+x=1+x+x22+x33+…x－x22+x33－x44+…=x+12x2+23x3+95x2+…+∑nk=0－1kk+1n－kxn+1+…x＜1。。2．6。64fx=tanxx=0。fx=tanxx=0tanx=A1x+A3x3+A5x5+…sinx=A1x+A3x3+A5x5+…cosxx－x33+x55+…=A1x+A3x3+A5x5+…1－x22+x44+…=A1x+A3－A12x3+A5+A14－A32x5+…A1=1A3－A12=－13A5+A14－A32=15…A1=1A3=13A5=215…tanx=x+13x3+215x5+…x＜π2。122．75。7fx=x2－8x+5x+2x－32x=0。fx=1x+2－2x－321x+2=12·11+x2=121－x2+x222－x323+…=12－x22+x223－x324+…x＜21x－3=－13·11－x3=－131+x3+x232－x333+…=－13－x32－x233－x334－…x＜31x－32=－ddx1x－3=－ddx∑∞n=0－xn3n+1=∑∞n=0ddxxn3n+1=∑∞n=1n3n+1xn－1=∑∞n=0n+13n+2xnx＜32x－32=29∑∞n=0n+13nxnx2－8x+5x+2x－32=1x+2－2x－32=518－43108x+11216x2－3713888x3+40923328x4－…x＜2。2．867。8fx=sin2xx=0。sin2x=121－cos2xcos2x=1－2x22+2x44－2x66+…=∑∞n=0－1n22n2nx2nsin2x=121－cos2x=∑∞n=0－1n－122n－12nx2n。、。2．9。9fx=e11－xx=0。fxx＜1f'x=e11－x·11－x21－x2·f'x－fx=01－x2·f''x+2x－3f'x=0…f0=ef'0=ef''0=3ef'''0=13e…e11－x=e1+x+32x2+133x3+…x＜1。2．10。10fx=excosxx=0。e1+ix=∑∞n=01+in·xnn=∑∞n=0槡2cosπ4+isinπ4n·xnn=∑∞n=02n2cosnπ4+isinnπ4·xnnx＜∞excosx=∑∞n=02n2cosnπ4xnnx＜∞exsinx=∑∞n=02n2sinnπ4·xnnx＜∞。3222J．2012351278－83．3．J．2011350522－25．4．J．201140320－22．5．J．200561114－17．6．D．2010．7．J．201103106－108．ＲesearchontheInfluenceofCampusFootballPromotiontotheDevelopmentofPrimarySchoolSportsMAOQi－xiaFacultyofPhysicalEducationBaotouTeachersCollegeBaotou014030AbstractThepresentsituationandexistingproblemsofthedevelopmentofcampusfootballinprimaryschoolsinBaotoucitywerestudiedbythemethodsofinvestigationinterviewandinduction．Thepromotionofcampusfoot-ballwasconducivetothedevelopmentofBaotouprimaryschoolphysicaleducation．Itwaslackofprofessionalsportsteachersandlessfootballfieldintheprocessofcampusfootballpromotion．KeywordsCampusfootballPromotionBaotoucity檶檶檶檶檶檶檶檶檶檶檶檶檶檶檶檶檶檶檶檶檶檶檶檶檶檶檶檶檶檶檶檶檶檶檶檶檶檶檶檶檶檶檶檶檶檶Primaryschoolsports22〔〕1．J．1998024－5．2．M．1980．3．J．200011341－352．4．M．．2000．5．M．．2001．6．M．2003．7．M．2005．8．J．199912046－7．SeveralExpansionMethodsofTaylorSeriesCAOQian－qianFacultyofMathematicsandStatisticsSuzhouUniversitySuzhou234000AbstractTaylorseriesisoneoftheimportantcontentsinhighermathematics．Butthegeneralteachingmate-rialsabouttheTaylorseriesexpansionmethodisnotdetailedenough．ThemethodofTaylorseriesexpansionandsomeexamplesarepresented．KeywordsFunctionTaylorseriesExpansionmethod23