Section Summary: 12.1 - Sequences

Definitions

Formally, a sequence is a function from the natural numbers ( ) into the reals ( ), , given explicitly (by showing the terms), by a formula, or by a graph. Informally, a sequence is a list of numbers written in a definite order: e.g. The numbers are called terms, and is called the term. Notice that it appears that our sequence just keeps on going: often our sequences will have terms corresponding to every natural number (the natural numbers themselves form a sequence!), and so are infinite in extent (countably infinite, like the natural numbers).

One famous sequence is the Fibonacci sequence, which is defined recursively by naming the first two terms in the sequence explicitly and then describing a pattern for the rest of the terms:

This sequence appears throughout nature, and was first described by a 13th century Italian mathematician.

We'll be interested especially in the behavior of sequences as the terms head off to infinity. A sequence has limit L and we write

if for every there is a corresponding integer N such that

If exists, then we say the sequence converges, or is convergent. Otherwise we say the sequence diverges, or is divergent.

Sometimes the sequence does not settle down to a limit, but diverges to : means that for every positive number M there is an integer N such that

A sequence is called increasing if for all (that is, . It is called decreasing if for all (that is, . It is called monotonic if it is either increasing or decreasing.

A sequence is called bounded above if there is a number M such that

It is called bounded below if there is a number m such that

If it is bounded both above and below, then it is a bounded sequence.

Theorems

If and when n is a natural number, then .

If and are convergent sequences and c is a constant, then

If for and , then .

If , then .

The sequence is convergent if and divergent for all other values of r: in particular,

Every bounded, monotonic sequence is convergent.

Properties, Hints, etc.

Summary

Sequences are like the natural numbers (1, 2, 3, .): they have distinct ordered terms traipsing off into the far distance. We're interested in what happens as the terms traipse off! Do they approach a fixed value? Do they oscillate, bouncing back and forth? Do they get larger and larger, or smaller and smaller? Several interesting example are included, such as the Fibonacci numbers (which came about from a rabbit population model!).

In this section we encounter many definitions, and a few theorems which help us to understand when a sequence converges (its terms approach a fixed value), or diverges (doesn't converge!). This is an issue of fundamental importance as we push on to our major objective: representing a function using an infinite sequence of functions! We start with numbers, of course, because that's a simpler case.

There are some interesting tricks which rely on our calculus background: using smooth functions to interpolate ``the data'' (the terms of the sequence), and limits of those functions at infinity.