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Double Series

  1. Sep 24, 2007 #1
    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 value analytically...any hints? :smile:

    Thanks!
     
    Last edited: Sep 24, 2007
  2. jcsd
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