Chapter 3

Sequences and CompactnessThis chapter is devoted to the fundamental properties of the real line which make all exis-tence theorems in Calculus possible. Of course you can follow stupid algorithms withoutthese things, but if you wish to understand what is going on, you need the concepts of thischapter.

3.1 SequencesFunctions defined on the set of integers larger than a given integer are called sequences.This turns out to be somewhat easier to consider in terms of limits than functions definedon Rwhich is why I am placing this early.

Definition 3.1.1 A function whose domain is defined as a set of the form

{k,k+1,k+2, · · ·}

for k an integer is known as a sequence. Thus you can consider

f (k) , f (k+1) , f (k+2) ,

etc. Usually the domain of the sequence is either N, the natural numbers consisting of{1,2,3, · · ·} or the nonnegative integers, {0,1,2,3, · · ·} . Also, it is traditional to writef1, f2, etc. instead of f (1) , f (2) , f (3) etc. when referring to sequences. In the abovecontext, fk is called the first term, fk+1 the second and so forth. It is also common to writethe sequence, not as f but as { fi}∞

i=k or just { fi} for short.

Example 3.1.2 Let {ak}∞

k=1 be defined by ak ≡ k2 +1.

This gives a sequence. In fact, a7 = a(7) = 72 +1 = 50 just from using the formula forthe kth term of the sequence.

It is nice when sequences come in this way from a formula for the kth term. However,this is often not the case. Sometimes sequences are defined recursively. This happens, whenthe first several terms of the sequence are given and then a rule is specified which deter-mines an+1 from knowledge of a1, · · · ,an. This rule which specifies an+1 from knowledgeof ak for k ≤ n is known as a recurrence relation.

Example 3.1.3 Let a1 = 1 and a2 = 1. Assuming a1, · · · ,an+1 are known, an+2 ≡ an+an+1.

Thus the first several terms of this sequence, listed in order, are 1, 1, 2, 3, 5, 8,· · · . Thisparticular sequence is called the Fibonacci sequence and is important in the study of repro-ducing rabbits. Note this defines a function without giving a formula for it. Such sequencesoccur naturally in the solution of differential equations using power series methods and inmany other situations of great importance.

3.2 Exercises1. Let g(t)≡

√2− t and let f (t) = 1

t . Find g◦ f . Include the domain of g◦ f .

2. Give the domains of the following functions.

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