微积分全英微积分全英 (1).pdf

9.1 Infinite SequencesProblem IntroductionThe area of circleGeometrical MeaningRArea of the 6-th1AArea of the 12-th2AArea of the -th162nnA,321nAAAASProblem IntroductionThe area of circleGeometrical Meaning12nDefinition of the Infinite SequenceDefinition:Infinite sequence1x2x3x4xnxA function whose domain is the set of positive integers and whose range is a set of real numbers.Explicit formula:Recursion formula:Eg:32,1nannEg:111,3,2nnaaanDefinition of the Infinite Sequence2,4,8,16,.2 31,.3 51,1,1,1,.21nn 1.nnn1 4+(1)2,2 3(1)n 2 n1(1)nnn 1234Definition of Infinite Sequencex1a2a2Na1Na3a2LLL|nnNaLIf for each positive number there is a corresponding positive number N such that（6）A sequence that fails to converge to any finite number is said to diverge,or to be divergent.Remark 2:For some fixed point ,written in some forms:x0 x0The sequence is said to converge to,and we write nalimnnaLExample 1Show that if is a positive integer,then 1lim0.pnn110ppnnAnalysis:1 .()pNn1 pnExample 1Show that if is a positive integer,then 1lim0.pnnLet an arbitrary 0 be given.Choose to be any number greater than 1.Then implies thatnaL10pn1=pn1pN11ppAnalysis:Show that if is a positive integer,then 1lim0.pnnTheorem A:Properties of Limits of SequencesLet and be convergent sequences and k be a constant.Then(i)(ii)(iii)(iv)(v)lim;nkklimlim;nnnnkakalimlimlim;nnnnnnnabablimlimlim;nnnnnnnabablimlim,provided that lim0.limnnnnnnnnnaabbbnanbExample 2Dose the sequence converge and,if so,to what number?Use the following almost obvious fact:If,then.limlimxnf xLf nLBy Hospitals Rule,lnlimxxxe1limxxxe=0Thus,So,lnconverges to 0.lnlim0nnnelnnneTheorem B:Squeeze TheoremSuppose that and both converge to L and that for (is a fixed integer).Then also converges to L.110,s.t.,(1)nnNnNaLLaL 22 ,s.t.,(2)nnNnNcLLcL12By ,Take max,nnnabcNN N so ,nnnbnN LacLwe have,namely,lim.nnnnbLbbLLProof:nancnbnnnabcnKKExample 3Show that ForSince lim10nnThe result follows by the Squeeze Theorem.3sinlim0.nnnlim 10nnand 31(sin,.1)1nnnnn Theorem CIfthen lim0,nnalim0.nna00nnnaaaExample 4Show that if then(1)If ,the result is trivial,so suppose otherwise.11limlim 10.nnnpnpBy the Binomial Formula,1nr1+nppnThus,10.nrpnlim0.nnr(2)Then ,1(positive terms)pn Since,and so for some positive number .11r 0r 1|1r 1|1rp 0p Example 4By the Squeeze Theorem,lim0.nnrlim0lim0.nnnnrrBy Theorem C,Show that if thenlim0.nnr11r Theorem D:Monotonic Sequence TheoremIf U is an upper bound for a non-decreasing sequence ,then the sequence converges to a limit A that is less than or equal to U.If L is a lower bound for a non-increasing sequence ,then the sequence converges to a limit B that is greater than or equal to L.12naaaU12nLbbbnab nSummaryDefinition of Infinite SequenceProperties of Limits of SequencesSqueeze TheoremTheorem CPay attention to grasp and consolidate the definition,the calculation,and the skilling in the application of theorems.Monotonic Sequence TheoremQuestions and Answers1Prove s1()is convergent with limit s12nnn 12nfor all n sufficiently large.11ln()ln2ln22ln2lnlnnnnn 1|1|,2nns Hence for any we need to show that0Questions and Answers1Proof s1()is convergent with limit s12 nnnso for any ,let,when ,0lnln2N nN1|1|.2nns we can getInfinite Sequences