**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.