Page 259 - Calculus Workbook For Dummies

Chapter 13

Infinite Series:

Welcome to the Outer Limits

In This Chapter

Twilight zone stuff
Serious series
Tests, tests, and more tests

n this chapter, you look at something that’s really quite amazing if you stop to think about
Iit: sums of numbers that never end. Seriously, the sums of numbers in this chapter — if
written out completely — would not fit in our universe. But despite the never-ending nature
of these sums, some of them add up to a finite number! These are called convergent series.
The rest are called divergent. Your task in this chapter is to decide which are which.

The Nifty nth Term Test

Because the mere beginning terms of any given sequence would completely fill the universe,
and because the nth term is way beyond that, where is it? Does it really exist or is it only a
figment of your imagination? If a tree falls in a forest and no one’s there to hear it, does it
make a sound?

First, a couple definitions. A sequence is a finite or infinite list of numbers (we will be dealing
only with infinite sequences). When you add up the terms of a sequence, the sequence
becomes a series. For example,

1, 2, 4, 8, 16, 32, 64, . . . is a sequence, and

1 + 2 + 4 + 8 + 16 + 32 + 64 + . . . is the related series.

If lim a n ! 0, then ! a n diverges. In English, this says that if a series’ underlying sequence
n " 3
does not converge to zero, then the series must diverge.
It does not follow that if a series’ underlying sequence converges to zero then the series will
definitely converge. It may converge, but there’s no guarantee.