极限的一个定理及其应用-外文翻译

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1、本科生毕业设计（论文）外文翻译学生姓名：张朋宇学号：408114010113专业班级：数学与应用数学指导教师：梁海燕老师2014年02月100ADiscussiononaLimitTheoremandItsApplicationAbstract：Thispaperproposesthatalimittheoremcanhelptosolveaspecificlimitproblemofsumformulaandthatsomelimitofproductformulacanalsobesolvedbyexploitingthef

2、eatureoflogarithmfunction.Keywords：limittheorem;sumformula;productformulaIncalculus,wewillusuallysolveaspecificlimitproblemofsumformulaZI5+1n+2:Butthissumformulacan’tsumdirectly，anditcan’tchangeintosomekindsoffunction’sintegralsum.Soitishardtoworkoutitslimit，forsolve

3、thisproblem.Thispaper’sproposesisthatalimittheoremcanhelptosolvethislimitproblemofsumformulaandthatsomelimitofproductformulacanalsobesolvedbylogarithmfunction.Theorem1Let(a)fbedifferentiableatx=0andf(0)=0,(b)gbeintegrableforfa,bl.Wehavenhg(^)^XkJ=f'(0)J^g(x)dx,whereT

4、:a=xIIrII=inax/^xx/ProofBythe(a),forevery£>0thereisa8^>0suchthatximplies

5、/(x)-fo)x

6、g(x)

7、forxE[a，b]andthereisa^2>0suchthatIITII<么implieshAv/-d.vLetS-min{

8、-/7o>jg(x)dx^1-f'(Q)fgfCMx,+1/70>

9、

10、-jg(x)d.r

11、^l/7o>

12、+f[s(^)-/7o>M/7o>

13、+gfQ•••fW为

14、gfQ么/

15、<^)II/7o>

16、+^A/yz4X/=7

17、/7o>

18、+M(b-a)]andthereforeMhlim^V//^g(^)^x，7=/70>J*gfx>dx.Wenotetheprecedingargumentwasbasedontheassumptionthatf(0)=0.Forthecasethatf(0)7^0.Wecanshowthatforf(

19、0)>0and1户'g(^)]=-ooforf(0)<0.Letf(x)=xthentheorem1hasbecomeVgfO〜'=(x)dx.Thisisdefinitionofdefiniteintegral，andbylogarithmfunctionwegetCorollary2Iffbedifferentiableatx=0andf(0)=1andgbeintegrableforxinto[a，b]thenwehaveInpracticalisusuallydivide[0,1]intonparts，andchoose

20、=—n(k=l，2,."，!!)•Corollary3Letfbedifferentiableatx=0andgbeintegrableforxinto[0,1],thenwehave(a)Iff(0)=0,wehave(b)(b)Iff(0)=1，wehaveProofBythattheorem1andlogarithmfunction，wegetlim^=limcxp

21、^=exp(1”'[S(^)=cp[[f(x)/^-oJg(x)d.rCr>dexpfVo>Jg(x)d.riExamplelEvaluateeacho

22、fthefollowing:Solution(a)Rewritethesumintheequivalentform2Lct/fx^^rand=x.Thenf'(Q)=0and丄2•Sothatbytheorem1，怕LT^+T2^…+7^nlim上n1+彳.tl