# Series

What is

What is

What is

Consider the following sequence of partial sums:

Can we compute

These partial sums look as follows:

It looks very likely that , if it makes any sense. But does it?

In a moment we will define

A little later we will show that , hence indeed .

Definition 6.2.1 (Sum of series)   If is a sequence, then the sum of the series is

provided the limit exists. Otherwise we say that diverges.

Example 6.2.2 (Geometric series)   Consider the geometric series for . Then

To see this, multiply both sides by and notice that all the terms in the middle cancel out. For what values of does converge? If , then and

If , then diverges, so diverges. If , it's clear since that the series also diverges (since the partial sums are ).

For example, if and , we get

as claimed earlier.

William Stein 2006-03-15