數(shù)學(xué)分析-高等數(shù)學(xué)-微積分-英語課件-chapter11b

1、The comparison testsTheorem 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.Ex. Determine whether converges.Sol. So the series converges.1hThe comparison testsTheorem SuThe lim

2、it comparison testTheorem Suppose that and are series withpositive terms. Suppose Then(i) when c is a finite number and c0, 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 of2hTh

3、e limit comparison testTheorExampleEx. Determine whether the following series converges.Sol. (1) diverge. choose then(2) diverge. take then(3) converge for p1 and diverge for take then3hExampleEx. Determine whether tQuestionEx. Determine whether the series converges or diverges.Sol. 4hQuestionEx. De

4、termine whether Alternating seriesAn alternating series is a series whose terms are alternatively positive and negative. For example,The n-th term of an alternating series is of the form where is a positive number.5hAlternating seriesAn alternatiThe alternating series testTheorem If the alternating

5、series satisfies (i) for all n (ii) Then the alternating series is convergent.Ex. The alternating harmonic series is convergent.6hThe alternating series testTheExampleEx. Determine whether the following series converges.Sol. (1) converge (2) convergeQuestion.7hExampleEx. Determine whether tAbsolute

6、convergenceA series is called absolutely convergent if the series of absolute values is convergent.For example, the series is absolutely convergent while the alternating harmonic series is not.A series is called conditionally convergent if it is convergent but not absolutely convergent.Theorem. If a

7、 series is absolutely convergent, then it is convergent.8hAbsolute convergenceA series ExampleEx. Determine whether the following series is convergent.Sol. (1) absolutely convergent (2) conditionally convergent 9hExampleEx. Determine whether tThe ratio testThe ratio test(1) If then is absolutely con

8、vergent.(2) If or then diverges.(3) If the ratio test is inconclusive: that is, noconclusion can be drawn about the convergence of10hThe ratio testThe ratio test10ExampleEx. Test the convergence of the seriesSol. (1) convergent (2) convergent for divergent for11hExampleEx. Test the convergencThe roo

9、t testThe root test(1) If then is absolutely convergent.(2) If or then diverges.(3) If the root test is inconclusive.12hThe root testThe root test12hExampleEx. Test the convergence of the seriesSol. convergent for divergent for13hExampleEx. Test the convergencRearrangementsIf we rearrange the order

10、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.By a rearrangement of an infinite series we mean a series obtained by simply changing the order of the terms.It turns out that: if is an absolutely convergent series wi

11、th sum , then any rearrangement of has the same sum .However, any conditionally convergent series can be rearranged to give a different sum.14hRearrangementsIf we rearrange Example Ex. Consider the alternating harmonic seriesMultiplying this series by we getorAdding these two series, we obtain15hExa

12、mple Ex. Consider the alterStrategy for testing seriesIf we can see at a glance that then divergenceIf a series is similar to a p-series, such as an algebraic form, or a form containing factorial, then use comparison test.For an alternating series, use alternating series test.16hStrategy for testing

13、 seriesIf Strategy for testing seriesIf n-th powers appear in the series, use root test.If f decreasing and positive, use integral test. Sol. (1) diverge (2) converge (3) diverge (4) converge17hStrategy for testing seriesIf Power seriesA power series is a series of the formwhere x is a variable and

14、are constants called coefficientsof series.For each fixed x, the power series is a usual series. We can test for convergence or divergence.A power series may converge for some values of x and diverge for other values of x. So the sum of the series is a function18hPower seriesA power series is Power

15、seriesFor example, the power seriesconverges to whenMore 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.19hPower seriesFor example, the pExampleEx. For what values of x is the power series convergent?Sol. By ratio test,the pow