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1、The limit 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 of第1页
2、/共21页ExampleEx.Determine whether the following series converges.Sol.(1)diverge.choose then(2)diverge.take then(3)converge for p1 and diverge for take then第2页/共21页QuestionEx.Determine whether the series converges or diverges.Sol.第3页/共21页Alternating seriesAn alternating series is a series whose terms
3、are alternatively positive and negative.For example,The n-th term of an alternating series is of the form where is a positive number.第4页/共21页The alternating series testTheorem If the alternating series satisfies(i)for all n (ii)Then the alternating series is convergent.Ex.The alternating harmonic se
4、ries is convergent.第5页/共21页ExampleEx.Determine whether the following series converges.Sol.(1)converge (2)convergeQuestion.第6页/共21页Absolute convergenceA series is called absolutely convergent if the series of absolute values is convergent.For example,the series is absolutely convergent while the alte
5、rnating harmonic series is not.A series is called conditionally convergent if it is convergent but not absolutely convergent.Theorem.If a series is absolutely convergent,then it is convergent.第7页/共21页ExampleEx.Determine whether the following series is convergent.Sol.(1)absolutely convergent (2)condi
6、tionally convergent 第8页/共21页The ratio testThe 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 of第9页/共21页ExampleEx.Test the convergence of the seriesSol.(1)convergent (2)convergent for di
7、vergent for第10页/共21页The root testThe root test(1)If then is absolutely convergent.(2)If or then diverges.(3)If the root test is inconclusive.第11页/共21页ExampleEx.Test the convergence of the seriesSol.convergent for divergent for第12页/共21页RearrangementsIf we rearrange the order of the term in a finite s
8、um,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 with sum ,then any rearrangeme
9、nt of has the same sum .However,any conditionally convergent series can be rearranged to give a different sum.第13页/共21页Example Ex.Consider the alternating harmonic seriesMultiplying this series by we getorAdding these two series,we obtain第14页/共21页Strategy for testing seriesIf we can see at a glance
10、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.第15页/共21页Strategy for testing seriesIf n-th powers appear in the series,use root test.If f decreasing and p
11、ositive,use integral test.Sol.(1)diverge (2)converge(3)diverge (4)converge第16页/共21页Power seriesA power series is a series of the formwhere x is a variable and are constants called coefficientsof series.For each fixed x,the power series is a usual series.We can test for convergence or divergence.A po
12、wer series may converge for some values of x and diverge for other values of x.So the sum of the series is a function第17页/共21页Power 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.第18页/共21页ExampleEx.For what values of x is the power series convergent?Sol.By ratio test,the power series diverges for all and only convergeswhen x=0.第19页/共21页Homework 24Section 11.4:24,31,32,42,46Section 11.5:14,34Section 11.6:5,13,23Section 11.7:7,8,10,15,36第20页/共21页感谢您的观看！第21页/共21页