math150b_9.1_9.6.pdf - Math 150B Jagodina Chapter 9.1-9.6...

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Math 150B, JagodinaChapter 9.1-9.6 PacketTutors may help.9.1 SequencesAsequence, denoted by{an}, is a function whose domain is restricted to numbersnZ+.We calla1, a2,· · ·, anthetermsof the sequence. In particular, we callan, thenthtermof the sequence.A sequence may be given by anexplicit formulaor by arecursiveformula.ExampleConsider the sequence 1,4,7,10,13,· · ·.Express the sequence explicitly andrecursively.ExampleFind the first six terms of the sequence defined by{an}=1-1n.What happens whenn→ ∞?1
Math 150B, JagodinaChapter 9.1-9.6 PacketTutors may help.DefinitionSuppose thatR. Thelimitof a sequence{an}is, denoted bylimn→∞an=means that for eachε >0, there existsM >0 such that ifn > M, then|an-|< ε. If theabove limit exists, we say the sequenceconvergesto, otherwise, we say that the sequencediverges.ExampleDetermine whether the given sequences converge or diverge. If it converges, proveit.1.an= (-1)n2.bn=1nTheoremSupposeR. Letf:RRsuch thatlimx→∞f(x) =‘.If{an}is a sequence such thatf(n) =anfor allnZ+, thenlimn→∞an=TheoremLetcR. Suppose that the sequences{an}and{bn}converge. Then,1.limn→∞(an±bn) = limn→∞an±limn→∞bn2.limn→∞can=chlimn→∞ani3.limn→∞(anbn) =hlimn→∞ani hlimn→∞bni4.limn→∞anbn=limn→∞anlimn→∞bnwherebn6= 0 and limn→∞bn6= 0 in number 4.ExampleFind the limit of each sequence or state that the sequence diverges1.an= 1-13n2.an= 3n-22
Math 150B, JagodinaChapter 9.1-9.6 PacketTutors may help.ExampleDetermine whether the sequences with the givennthterms converge or diverge.1.an=n2+ 13n2+n2.bn=2n5n3
Math 150B, JagodinaChapter 9.1-9.6 PacketTutors may help.DefinitionLetnN. Then,nfactorial, denoted byn!, is the productn! = 1·2·3· · ·(n-1)·nwith 0! = 1.ExampleEvaluate.1. 3n!2. (3n)!3.(n+ 2)!n!4
Math 150B, JagodinaChapter 9.1-9.6 PacketTutors may help.Theorem[Squeeze Theorem for Sequences] Iflimn→∞an== limn→∞bnand there exists a numberNZ+such thatancnbnfor alln > N, thenlimn→∞cn=ExampleShow that the sequence{cn}=ncosnnoconverges, and find its limit.5
Math 150B, JagodinaChapter 9.1-9.6 PacketTutors may help.TheoremLet{an}be a sequence. Iflimn→∞|an|= 0then,limn→∞an= 0.ExampleShow that the sequence{an}=(-1)n1nconverges, and find its limit.Pattern Recognition for SequencesExampleFor each sequence, write an expression for thenth term:1.

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