微积分讲义chap.pdf

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1、Chapter 2 Sequences in RChapter 2 Sequences in R2.1.Limits of sequences2.1.Limits of sequencesDef:A sequence(infinite sequence)is a function from N N into R R,xn=f(n),wedenote it by xn orxnnNnN.N.2.1.Definition:Let xn be a sequence of R R,xn is said to convergent to aR Rif for every0,NN N s.tWe deno

2、te it bylimxn=anxn-aN.1Example.Show that is convergentnProof:We showlimnGiven 0.1=0.n1-0N.nTo find anNN N s.tBy Archimedean principleNN Ns.tN1.1Nfor nN,we have110To findNN N s.txn1 11.TakeN=1Thenxn-1=0N.limxn=1nExercise.Show that 1 is convergent2n2.3 Ex:ampleShow(-1)n has no limit.Proof:supposelim(-

3、1)n=a.nGiven=1NN Ns.t21(-1)n-aN.2Let nN be even,then n+1 is odd.2=1n-(-1)n1=(1na)(1)n1a(1)n a (1)n1 a0a=b.Proof:Suppose a b.Let L=a b0 and let=L=a b0.liman=L1andliman=L2nnN1N Ns.tan L12for n N1similarly,N2N Ns.tan L2For nmax(N1,N2)We havean L1for nmax(N1,N2)2for n N22andan L22L1 L2 L1an an L2 L1an L

4、2an22Sinceis arbitrary,we derive thatL1=L2.2.5,Definition.By a subsequence of a sequence xn,we shall mean a sequence xnk wheren1n2n3.k.lim xn Ln N N Ns.txn L for nN.Letk Nn1 n2 n3kxnk L for nkk.limxnk=Lnk2.7.Definition.Let xn be a sequence of R R,xn is called bounded ifM0s.t,xnM for each n.2.8.TheoremEvery convergent sequence of R R is boundedProof:Let xn converge to LTo show xn is bounded.We must find a M s.txnMlim xn LnGiven1,NN Ns.txn L 1for nNxnNLet M=maxx1,x2,x3xN,L+1It is obvious,xnMnN Nxn is bounded