(adsbygoogle = window.adsbygoogle || []).push({}); 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:

[tex]\sum_{m,n=2}^\infty \frac{1}{m^n-1}[/tex]

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:

[tex]\sum_{m=2}^\infty \left( \sum_{n=2}^\infty \frac{1}{m^n-1} \right)[/tex]

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

[tex]\sum_{n=2}^\infty \frac{1}{m^n}[/tex]

which is convergent, since

[tex]\frac{1}{m}<1[/tex]

Thus:

[tex]\sum_{n=2}^\infty \frac{1}{m^n-1}[/tex] converges.

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

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

Thanks!

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# Homework Help: Double Series

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