# Double Series

1. Sep 24, 2007

### Emanuel84

1. The problem statement, all variables and given/known data

Determine the character of the following double series, eventually find its sum in analytical form:

$$\sum_{m,n=2}^\infty \frac{1}{m^n-1}$$

2. The attempt at a solution

At first look, this series can only be either convergent or divergent (i.e. regular), since it's a positive term series.
I rewrote it in a more convenient form in order to visualize it better:

$$\sum_{m=2}^\infty \left( \sum_{n=2}^\infty \frac{1}{m^n-1} \right)$$

At this point, I thought to use the comparison test for the series within the parenthesis, being the test series the geometric series:

$$\sum_{n=2}^\infty \frac{1}{m^n}$$

which is convergent, since

$$\frac{1}{m}<1$$

Thus:

$$\sum_{n=2}^\infty \frac{1}{m^n-1}$$ converges.

Iterating this procedure, I found that the whole double series converges.

The problem, now, is to find its value analytically...any hints?

Thanks!

Last edited: Sep 24, 2007