The Art and Craft of Problem Solving

346 CHAPTER 9 CALCULUS

other words, complete "local" information yields complete global information! This

is worth pondering.

In practice, it is not always necessary to use (8). As long as you know (or suspect)

that the series exists, you can opportunistically extract terms of a series.

Expand - 2 _

1

- into a power series about x = o.

x + 1

Solution: We simply use the geometric series tool (see page 133):

1

and thus

(^1 246)

--= 1 - x +x -x + ....

x^2 + 1

Example 9.4.4 Expand � into a power series about x = o.

Solution: Just substitute t = x^2 into the familiar series

t^2 t^3

et = l+t +-+-+···

2! 3!

You may wonder about these last two examples, asking, "Yes, we got a power

series, but how do we know that we actually got the Taylor series that we would have

gotten from (8)?" Once again, don't worry, for the power series expansion is unique.

The essential reason is just a generalization of the "derivative is the best linear approxi­

mation" idea mentioned on page 330. For example, let P 2 (X) denote the second-degree

Taylor polynomial for f(x) about x = a. We claim that P 2 (X) is the best quadratic

approximation to f(x) for the simple reason that

lim

f(x) -P 2 (X)

= 0,

x----> a (x - a)^2

while if Q(x) -I-P 2 (X) is any other quadratic polynomial, then

lim

f(x) - Q(x)

-I-O.

x ----> a (x - a?

(9)

Thus P 2 (X) is the only quadratic function that agrees with f(x),f'(x), and f"(x) when

x=a.

This is one reason why power series are so important. Not only are they easy to

manipulate, but they provide "ideal" information about the way functions grow.

Eulerian Mathematics

In the last few pages, we have been deliberately cavalier about rigor, partly because

the technical issues involved are quite difficult, but mostly because we feel that too

much attention to rigor and technical issues can inhibit creative thinking, especially at

two times: