数学分析-高等数学-微积分-英语课件--chapter11b.ppt

The comparison testsnTheorem Suppose that and are series with positive terms, then(i) If is convergent and for all n, then is also convergent.(ii) If is divergent and for all n, then is also divergent.nEx. Determine whether converges.nSol. So the series converges.1•hThe comparison testsTheorem SuThe limit comparison testnTheorem Suppose that and are series withpositive terms. Suppose Then(i) when c is a finite number and c>0, then either both series converge or both diverge.(ii) when c=0, then the convergence of implies the convergence of(iii) when then the divergence of implies thedivergence of2•hThe limit comparison testTheorExamplenEx. Determine whether the following series converges.nSol. (1) diverge. choose then(2) diverge. take then(3) converge for p>1 and diverge for take then3•hExampleEx. Determine whether tQuestionnEx. Determine whether the series converges or diverges.nSol. 4•hQuestionEx. Determine whether Alternating seriesnAn alternating series is a series whose terms are alternatively positive and negative. For example,nThe n-th term of an alternating series is of the form where is a positive number.5•hAlternating seriesAn alternatiThe alternating series testnTheorem If the alternating series satisfies (i) for all n (ii) Then the alternating series is convergent.nEx. The alternating harmonic series is convergent.6•hThe alternating series testTheExamplenEx. Determine whether the following series converges.nSol. (1) converge (2) convergenQuestion.7•hExampleEx. Determine whether tAbsolute convergencenA series is called absolutely convergent if the series of absolute values is convergent.nFor example, the series is absolutely convergent while the alternating harmonic series is not.nA series is called conditionally convergent if it is convergent but not absolutely convergent.nTheorem. If a series is absolutely convergent, then it is convergent.8•hAbsolute convergenceA series ExamplenEx. Determine whether the following series is convergent.nSol. (1) absolutely convergent (2) conditionally convergent 9•hExampleEx. Determine whether tThe ratio testnThe ratio test(1) If then is absolutely convergent.(2) If or then diverges.(3) If the ratio test is inconclusive: that is, noconclusion can be drawn about the convergence of10•hThe ratio testThe ratio test10ExamplenEx. Test the convergence of the seriesnSol. (1) convergent (2) convergent for divergent for11•hExampleEx. Test the convergencThe root testnThe root test(1) If then is absolutely convergent.(2) If or then diverges.(3) If the root test is inconclusive.12•hThe root testThe root test12hExamplenEx. Test the convergence of the seriesnSol. convergent for divergent for13•hExampleEx. Test the convergencRearrangementsnIf we rearrange the order of the term in a finite sum, then of course the value of the sum remains unchanged. But this is not the case for an infinite series.nBy a rearrangement of an infinite series we mean a series obtained by simply changing the order of the terms.nIt turns out that: if is an absolutely convergent series with sum , then any rearrangement of has the same sum .nHowever, any conditionally convergent series can be rearranged to give a different sum.14•hRearrangementsIf we rearrange Example nEx. Consider the alternating harmonic seriesMultiplying this series by we getorAdding these two series, we obtain15•hExample Ex. Consider the alterStrategy for testing seriesnIf we can see at a glance that then divergencenIf a series is similar to a p-series, such as an algebraic form, or a form containing factorial, then use comparison test.nFor an alternating series, use alternating series test.16•hStrategy for testing seriesIf Strategy for testing seriesnIf n-th powers appear in the series, use root test.nIf f decreasing and positive, use integral test. nSol. (1) diverge (2) converge (3) diverge (4) converge17•hStrategy for testing seriesIf Power seriesnA power series is a series of the formwhere x is a variable and are constants called coefficientsof series.nFor each fixed x, the power series is a usual series. We can test for convergence or divergence.nA power series may converge for some values of x and diverge for other values of x. So the sum of the series is a function18•hPower seriesA power series is Power seriesnFor example, the power seriesconverges to whennMore generally, A series of the formis called a power series in (x-a) or a power series centeredat a or a power series about a.19•hPower seriesFor example, the pExamplenEx. For what values of x is the power series convergent?nSol. By ratio test,the power series diverges for all and only convergeswhen x=0.20•hExampleEx. For what values of Homework 24nSection 11.4: 24, 31, 32, 42, 46nSection 11.5: 14, 34nSection 11.6: 5, 13, 23nSection 11.7: 7, 8, 10, 15, 3621•hHomework 24Section 11.4: 24, 3。