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Thread: Using Power Series to find this sum

  1. #1
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    Using Power Series to find this sum

    Can anyone help me on this? I'm trying to find the sum of

    $\displaystyle \sum_1^\infty \frac{k}{3^{2k}}$ using power series.

    This looks like a geometric sequence to me, but I can't get it in a form that lets me use a power series... any help?
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  2. #2
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    $\displaystyle \sum\limits_{k = 1}^\infty {x^k } = \frac{x}{{1 - x}}$
    Differenate to get $\displaystyle \sum\limits_{k = 1}^\infty {kx^{k-1} } = \frac{1}{{(1 - x)^2}}$
    Multiply by x $\displaystyle \sum\limits_{k = 1}^\infty {kx^{k} } = \frac{x}{{(1 - x)^2}}$.
    In your problem let $\displaystyle x=\frac{1}{9}$.
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