文档介绍：Chapter Ten
Sequences, Series, and All That
Introduction
Suppose we want pute an approximation of the number e by using the Taylor
x
polynomial pn for f (x) = e at a =0. This polynomial is easily seen to be
x 2 x 3 x n
pn ( x) = 1 + x + + +K+ .
2 6 n!
We could now use pn (1) as an approximation to e . We know from the previous chapter
that the error is given by
e x
e - p (1) = 1n+1 ,
n (n + 1)!
where 0 < x <1 . Assume we know that e <3, and we have the estimate
3
0 £ e - p (1) £ .
n (n +1)!
Meditate on this error estimate. It tells us that we can make this error as small as we like by
choosing n sufficiently large. This is expressed formally by saying that the limit of
pn (1) as n es infinite is e . This is the idea we shall study in this chapter.
Sequences
A sequence of real numbers is simply a function from a subset of the nonnegative
integers into the reals. If the domain is infinite, we say the sequence is an infinite
sequence. (Guess what a finite sequence is.) We shall be concerned only with infinite
sequences, and so the modifier will usually be omitted. We shall also almost always
consider sequences in which the domain is either the entire set of nonnegative or positive
integers.
There are several notational conventions involved in writing and talking about
sequences. If f :Z+ ® R , it is customary to denote f (n) by f n , and the sequence itself
æ1ö
by ( f ) . (Here Z denotes the positive integers.) Thus, for example, ç ÷ is the sequence
n + ènø
1
f defined by f (n) = . The function values f are called terms of the sequence.
n n
Frequently one sees a sequence described by writing something like
2
14, ,9,K,n ,K .
This is simply another way of describing the sequence (n 2 ).
Let (an ) be a sequence and suppose there is a number L such that for any e >0,
there is an integer N such that |an - L|< e for all n > N . Then L is said to be a limit of the
sequence, and (an ) is said to co