The Art and Craft of Problem Solving

344 CHAPTER 9 CALCULUS

• If fn(x) ---t f(x) unifonnly, then J fn (x)dx ---t J f(x)dx.


• Unifonnly convergent power series can be differentiated and integrated tenn by
  tenn.
  The last item is important. A power series is a special case of a series of functions,
  namely one where each tenn has the fonn an (x -c)n. In your elementary calculus
  courses, you learned about the radius of convergence. II For example, the series

l+x+�+x^3 + ...

converges to _

1

_, provided that Ixl < 1. In other words, the values of x for which the

I-x

series converge lie within 1 unit of O. The center of convergence is 0 and the radius

of convergence is I. What makes power series so useful is the fact that they converge

unifonnly as long as you contract the radius of convergence a bit. More fonnally,

Let

ao+alx+a 2 �+'" =f(x)

fo r all x such that Ix -cl < R. Thenfor any positive e, the convergence

is uniform fo r all x such that Ix -cl :s; R -e.

Thus, once you are in possession of a unifonnly convergent power series, you can

abuse it quite a bit without fear of mathematical repercussions. You can differentiate

or integrate tenn by tenn, multiply it by other well-behaving power series, etc., and be

sure that what you get will behave as you think it should.

Taylor Series with Remainder

Most calculus textbooks present Taylor series, but the proof is rarely mentioned, or is

relegated to a technical appendix that is never read. This is a shame, because it is as

easy as it is important. Let us derive the familiar Taylor series fonnula (including the

remainder tenn) in a way that is both easy to understand and remember, with a simple

example.

Example 9.4.2 Find the second-degree Taylor polynomial for f(x) , plus the remain­

der.

Solution: Assume that f(x) is infinitely differentiable on its domain D, and all

derivatives are bounded. In other words, for each k � 1 there is a positive number Mk

such that If(k)(x)1 :s; Mk for all x in the domain. We shall construct the second-degree

Taylor polynomial about x = a (where a E D). To do this, we start with the third

derivative. All that we know for sure is that

• M 3 < f"'(t) < M 3

(^11) To really understand radius of convergence, you need to look at the complex plane. See [29] for an illumi­
nating discussion.