《数学分析高等数学微积分英语上海交通大学学习教案.pptx》由会员分享，可在线阅读，更多相关《数学分析高等数学微积分英语上海交通大学学习教案.pptx（21页珍藏版）》请在taowenge.com淘文阁网|工程机械CAD图纸|机械工程制图|CAD装配图下载|SolidWorks_CaTia_CAD_UG_PROE_设计图分享下载上搜索。
1、会计学1数学分析数学分析(sh xu fn x)高等数学微积分高等数学微积分英语上海交通大学英语上海交通大学第一页，共21页。The limit comparison testn nTheorem Theorem Suppose that and are series withSuppose that and are series withpositive terms.Suppose positive terms.Suppose ThenThen(i)when(i)when c c is a finite number and is a finite number and cc0,then e
2、ither both series 0,then either both series converge or both diverge.converge or both diverge.(ii)when(ii)when c c=0,then the convergence of implies the=0,then the convergence of implies the convergence ofconvergence of(iii)when then the divergence of implies the(iii)when then the divergence of impl
3、ies thedivergence ofdivergence of第1页/共21页第二页，共21页。Examplen nEx.Ex.Determine whether the following series converges.Determine whether the following series converges.n nSolSol.(1)diverge.choose then.(1)diverge.choose then(2)diverge.take then(2)diverge.take then(3)converge for(3)converge for pp1 and di
4、verge for take1 and diverge for take then then第2页/共21页第三页，共21页。Questionn nEx.Ex.Determine whether the series Determine whether the series converges or diverges.converges or diverges.n nSolSol.第3页/共21页第四页，共21页。Alternating seriesn nAn An alternating seriesalternating series is a series whose terms are
5、 alternatively positive and negative.For is a series whose terms are alternatively positive and negative.For example,example,n nThe The n n-th term of an alternating series is of the form-th term of an alternating series is of the form where is a positive number.where is a positive number.第4页/共21页第五
6、页，共21页。The alternating series testn nTheorem Theorem If the alternating seriesIf the alternating series satisfies(i)for all satisfies(i)for all n n (ii)(ii)Then the alternating series is convergent.Then the alternating series is convergent.n nExEx.The.The alternating harmonic seriesalternating harmo
7、nic series is convergent.is convergent.第5页/共21页第六页，共21页。Examplen nExEx.Determine whether the following series converges.Determine whether the following series converges.n nSolSol.(1)converge (2)converge.(1)converge (2)convergen nQuestionQuestion.第6页/共21页第七页，共21页。Absolute convergencen nA A series is
8、called series is called absolutely convergentabsolutely convergent if the series of absolute values is if the series of absolute values is convergent.convergent.n nFor example,the series is absolutely convergentFor example,the series is absolutely convergent while the alternating harmonic series is
9、not.while the alternating harmonic series is not.n nA A series is called series is called conditionally convergentconditionally convergent if it is convergent but not absolutely if it is convergent but not absolutely convergent.convergent.n nTheoremTheorem.If a series is absolutely convergent,then i
10、t is convergent.If a series is absolutely convergent,then it is convergent.第7页/共21页第八页，共21页。Examplen nExEx.Determine whether the following series is convergent.Determine whether the following series is convergent.n nSolSol.(1)absolutely convergent.(1)absolutely convergent (2)conditionally convergent
11、 (2)conditionally convergent 第8页/共21页第九页，共21页。The ratio testn nThe ratio testThe ratio test(1)If then is absolutely convergent.(1)If then is absolutely convergent.(2)If or then diverges.(2)If or then diverges.(3)If the ratio test is inconclusive:that is,no(3)If the ratio test is inconclusive:that is
12、,noconclusion can be drawn about the convergence ofconclusion can be drawn about the convergence of第9页/共21页第十页，共21页。Examplen nExEx.Test the convergence of the series.Test the convergence of the seriesn nSolSol.(1)convergent.(1)convergent (2)convergent for divergent for (2)convergent for divergent fo
13、r第10页/共21页第十一页，共21页。The root testn nThe root testThe root test(1)If then is absolutely convergent.(1)If then is absolutely convergent.(2)If or then diverges.(2)If or then diverges.(3)If the root test is inconclusive.(3)If the root test is inconclusive.第11页/共21页第十二页，共21页。Examplen nExEx.Test the conve
14、rgence of the series.Test the convergence of the seriesn nSolSol.convergent for divergent for convergent for divergent for第12页/共21页第十三页，共21页。Rearrangementsn nIf we rearrange the order of the term in a If we rearrange the order of the term in a finitefinite sum,then of course the value of the sum sum
15、,then of course the value of the sum remains unchanged.But this is not the case for an infinite series.remains unchanged.But this is not the case for an infinite series.n nBy a By a rearrangementrearrangement of an infinite series we mean a series obtained by simply of an infinite series we mean a s
16、eries obtained by simply changing the order of the terms.changing the order of the terms.n nIt turns out that:if is an absolutely convergent series with sum ,then any It turns out that:if is an absolutely convergent series with sum ,then any rearrangement of has the same sum .rearrangement of has th
17、e same sum .n nHowever,any conditionally convergent series can be rearranged to give a different sum.However,any conditionally convergent series can be rearranged to give a different sum.第13页/共21页第十四页，共21页。Example n nExEx.Consider the alternating harmonic series.Consider the alternating harmonic ser
18、iesMultiplying this series by we getMultiplying this series by we getororAdding these two series,we obtainAdding these two series,we obtain第14页/共21页第十五页，共21页。Strategy for testing seriesn nIf we can see at a glance that then divergenceIf we can see at a glance that then divergencen nIf a series is si
19、milar to a If a series is similar to a p p-series,such as an algebraic form,or a form containing factorial,-series,such as an algebraic form,or a form containing factorial,then use comparison test.then use comparison test.n nFor an alternating series,use alternating series test.For an alternating se
20、ries,use alternating series test.第15页/共21页第十六页，共21页。Strategy for testing seriesn nIf If n n-th powers appear in the series,use root test.-th powers appear in the series,use root test.n nIf If f f decreasing and positive,use integral test.decreasing and positive,use integral test.n nSolSol.(1)diverge
21、 (2)converge(3)diverge (4)converge.(1)diverge (2)converge(3)diverge (4)converge第16页/共21页第十七页，共21页。Power seriesn nA A power seriespower series is a series of the form is a series of the formwhere where x x is a variable and are constants called coefficients is a variable and are constants called coef
22、ficientsof series.of series.n nFor each fixed For each fixed x x,the power series is a usual series.We can test for,the power series is a usual series.We can test for convergence or divergence.convergence or divergence.n nA power series may converge for some values of A power series may converge for
23、 some values of x x and diverge for and diverge for other values of other values of x x.So the sum of the series is a function.So the sum of the series is a function第17页/共21页第十八页，共21页。Power seriesn nFor example,the power seriesFor example,the power seriesconverges to whenconverges to whenn nMore gen
24、erally,A series of the formMore generally,A series of the formis called a is called a power series in(power series in(x x-a a)or a or a power series centeredpower series centeredat at a a or a or a power series about power series about a a.第18页/共21页第十九页，共21页。Examplen nExEx.For what values of.For wha
25、t values of x x is the power series is the power series convergent?convergent?n nSolSol.By ratio test,.By ratio test,the power series diverges for all and only convergesthe power series diverges for all and only convergeswhen when x x=0.=0.第19页/共21页第二十页，共21页。Homework 24n nSection 11.4:24,31,32,42,46n nSection 11.5:14,34n nSection 11.6:5,13,23n nSection 11.7:7,8,10,15,36第20页/共21页第二十一页，共21页。