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Sequences: Convergence and Divergence

In Section2.1, we consider (infinite) sequences, limits of sequences, and bounded and monotonic sequences of real numbers. In addition to certain basic properties of convergent sequences, we also study divergent sequences and in particular, sequences that tend to positive or negative infinity. We present a number of methods to discuss convergent sequences together with techniques for calculating their limits. Also, we prove thebounded monotone convergence theorem(BMCT), which asserts that every bounded monotone sequence is convergent. In Section2.2, we define the limit superior and the limit inferior. We continue the discussion with Cauchy sequences and give ex- amples of sequences of rational numbers converging to irrational numbers. As applications, a number of examples and exercises are presented.2.1 Sequences and Their Limits An infinite(real) sequence(more briefly, a sequence) is a nonterminating collection of (real) numbers consisting of a first number, a second number, a third number, and so on: a1 ,a 2 ,a 3

Specifically, ifnis a positive integer, thenan

is called thenth term of the sequence, and the sequence is denoted by {a 1 ,a 2 ,...,a n ,...}or, more simply,{a n For example, the expression{2n}denotes the sequence 2,4,6,....Thus,a sequence of real numbers is a special kind of function, one whose domain is the set of all positive integers or possibly a set of the form{n:n≥ k}for some fixedk?Z, and the range is a subset ofR. Let us now make this point precise.

Definition 2.1.Arealsequence{an

}is a real-valued functionfdefined on aset{ k,k+1,k+2,...}. The functional valuesS. Ponnusamy,Foundations of Mathematical Analysis,

DOI10.1007/978-0-8176-8292-7

2,

©Springer Science+Business Media, LLC 201223

24 2 Sequences: Convergence and Divergence

f(k),f(k+1),f(k+2),... are called thetermsof the sequence. It is customary to writef(n)=a n for n≥k, so that we can denote the sequence by listing its terms in order; thus we write a sequence as {a n n≥k or{a n+k-1 ∞n=1 or{a n ∞n=k or{a k ,a k+1

The numbera

n is called thegeneral termof the sequence{a n }(nth term, especially fork=1).Theset{a n :n≥k}is called therange of the sequence {a n n≥k . Sequences most often begin withn=0orn=1,inwhichcase the sequence is a function whose domain is the set of nonnegative integers (respectively positive integers). Simple examples of sequences are the se- quences of positive integers, i.e., the sequence{a n }for whicha n =nfor n≥1,{1/n},{(-1) n },{(-1) n +1/n}, and the constant sequences for which a n =cfor alln.TheFibonacci sequenceis given by a 0 ,a 1 =1,a 2 =2,a n =a n-1 +a n-2 forn≥3. The terms of this Fibonacci sequence are calledFibonacci numbers,andthe first few terms are

1,1,2,3,5,8,13,21.

2.1.1 Limits of Sequences of Real Numbers

A fundamental question about a sequence{a

n }concerns the behavior of its nth terma n asngets larger and larger. For example, consider the sequence whose general term is a n =n+1 n=1+1n. It appears that the terms of this sequence are getting closer and closer to the number 1. In general, if the terms of a sequence can be made as close as we please to a numberafornsufficiently large, then we say that the sequence converges toa.Here is a precise definition that describes the behavior of a sequence.

Definition 2.2 (Limit of a sequence).Let{a

n }be a sequence of real num- bers. We say that the sequence{a n }converges to the real numbera,ortends toa, and we write a= lim n→∞ a n or simplya= lima n if for every?>0, there is an integerNsuch that |a n -a|2.1 Sequences and Their Limits 25 In this case, we call the numberaa limit of the sequence{a n }. We say that the sequence{a n }converges (or is convergent or has limit) if it converges to some numbera. A sequence diverges (or is divergent) if it does not converge to any number. For instance, in our example above we would expect lim n→∞ n+1 n=1. The notions of convergence and limit of a sequence play a fundamental role in analysis.

Ifa?R, other notations for the convergence of{a

n }toaare lim n→∞ (a n -a)=0 anda n →aasn→∞.

The notationa= lima

n means thateventuallythe terms of the sequence{a n can be made as close toaas may be desired by takingnsufficiently large.

Note also that

|a n -a|That is, a sequence{a n }converges toaif and only if every neighborhood ofa contains all but a finite number of terms of the sequence. SinceNdepends on ?, sometimes it is important to emphasize this and writeN(?) instead ofN. Note also that the definition requires someN, but not necessarily the smallest Nthat works. In fact, if convergence works for someNthen anyN 1 >Nalso works.

To motivate the definition, we again considera

n =(n+1)/n.Given?>0, we notice that????n+1 n-1????=1n1?. Thus,Nshould be some natural number larger than 1/?. For example, if ?=1/99, then we may chooseNto be any positive integer bigger than 99, and we conclude that????n+1 n-1????Thus,Nclearly depends on?. The definition of limit makes it clear that changing a finite number of terms of a given sequence affects neither the convergence nor the divergence of the sequence. Also, we remark that the number?provides a quantitative measure of "closeness," and the numberNa quantitative measure of "largeness." We now continue our discussion with a fundamental question:Is it possible for a sequence to converge to more than one limit?quotesdbs_dbs30.pdfusesText_36

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