微積分英文課件：chapter8.4 Series of Nonnegative Terms

1、8.4 Series of Nonnegative TermsnSSS210,nnaaA series with nonnegative terms is convergent iff the sequences of its partial sums is bounded.Series of Nonnegative TermsnaCorollary of Theorem 5222222211111111121231ndxnnx11111111123ndxnnxFor exampleConvergesDivergesTheorem:Proof of The Integral Test:(4)(

2、5)111( )nninnif x dxaasExampleTest the series for convergence or divergence.21nnnesolution2( )xxf xeBecause and222(1 2 )( )0()xxe xxf xewhen1x,so the function is continuous,2( )xxf xepositive,and decreating on . 1,We use the integral test: . 2222111lim1111limlim=2222txxtxttttxxdxdxeeeeeeThus, is con

3、vergent integral and21xxdxeso,by the integral test,the series21nnneis convergent.Example:Converges Diverges Note:Note:The Comparison Testproof(i) letniinasniinbt1nnbtSince both series have positive terms, ,iiba the sequences and are increasing. ns ntIf is convergent then for all n, 1nnbttnSince we h

4、avettsnnfor all n.This means that is incrasing and bounded and therefore converges by the Monotonic Sequence Theorem.thus, converges.1nna ns(ii)If is convergent then, 1nnbnt,iiba Since we havennts for all n.Thus,.ns1nnaTherefore diverges.Most of the time we use a p-series or a geometricseries for th

5、e purpose of comparisonExample Determine whether the series converges Or diverges: 1)1(1).1(nnn 1)4)(1(1).2(nnnSolution)1(1).1( nnun2)1(1 n,11 n 111nndiverges)4)(1(1).2( nnunnn 1,12n 11(1)nson ndiverges 121nnconverges11(1)(4)nsonnconvergesExample Determine whether the series converges Or diverges:14

6、25nnSolution:44255nnconverges145nn1425nnsoconvergesExample: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 term

7、s and suppose thatna1nnnalimaThen(a)the series converges if(b)the series deverges if or is infinite(c)the test is inconclusive if111Proof(a) 1. 1 rLetnnnaa1limraann1NNraa1Nn whenSinceThusThat isNNNarraa212NmmNmNarraa1Example:Solution:Example:Investigate the convergence of the following series;)!1(1).1 (1nn1!2).3(nnnnn;10!).2(1nnn;2tan ).4(11nnnThe nth-Root TestLet be a series with for and suppose thatnannnlimaThen(a)the series converges if(b)the series deverges if or is infinite(c)t