## The Ratio Test

Recall that is a geometric series if and only if for some fixed  and . Here we call  the common ratio. Notice that the ratio of any two successive terms is :

Moreover, we have converges (to ) if and only if (and, of course it diverges if ).

Example 6.4.5   For example, converges to . However, diverges.

Theorem 6.4.6 (Ratio Test)   Consider a sum . Then
1. If then is absolutely convergent.
2. If then diverges.
3. If then we may conclude nothing from this!

Proof. We will only prove 1. Assume that we have . Let , and notice that (since , so , so , and also ).

Since , there is an  such that for all we have

so

Then we have

Here the common ratio for the second one is , hence thus the right-hand series converges, so the left-hand series converges.

Example 6.4.7   Consider . The ratio of successive terms is

Thus this series converges absolutely. Note, the minus sign is missing above since in the ratio test we take the limit of the absolute values.

Example 6.4.8   Consider . We have

Thus our series diverges. (Note here that we use that .)

Example 6.4.9   Let's apply the ratio test to . We have

This tells us nothing. If this happens... do something else! E.g., in this case, use the integral test.

William Stein 2006-03-15