微积分教学资料——chapter8教学课件

A series with nonnegative terms is convergent iff the sequences of its partial sums is bounded.Series of Nonnegative TermsCorollary of Theorem 5For exampleConvergesDivergesTheorem:Proof of The Integral Test:(4)(5)ExampleTest the series for convergence or divergence.solutionBecause andwhenso the function is continuous,positive,and decreating on .We use the integral test:.Thus,is convergent integral andso,by the integral test,the seriesis convergent.Example:Converges Diverges Note:Note:The Comparison Testproof(i)letSince both series have positive terms,the sequences and are increasing.If is convergent then for all n,Since we havefor all n.This means that is incrasing and bounded and therefore converges by the Monotonic Sequence Theorem.thus,converges.(ii)If is convergent then,Since we havefor all n.Thus,Therefore diverges.Most of the time we use a p-series or a geometricseries for the purpose of comparisonExample Determine whether the series converges Or diverges:SolutiondivergesdivergesconvergesconvergesExample Determine whether the series converges Or diverges:Solution:convergesconvergesExample:Solution:andconvergesExample:Solution:andTheorem:Example:then:and So the given series converges by the limit comparison test.Example:So the given series converges by the limit comparison test.The Ratio TestLet be a series with positiv terms and suppose thatThen(a)the series converges if(b)the series deverges if or is infinite(c)the test is inconclusive ifProof(a)LetwhenSinceThusThat isExample:Solution:Example:Investigate the convergence of the following seriesThe nth-Root TestLet be a series with for and suppose thatThen(a)the series converges if(b)the series deverges if or is infinite(c)the test is inconclusive ifExample:Solution:Investigate the convergence of the following seriesExample:谢谢