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Math Help - Finding sum of infinite series

  1. #1
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    Finding sum of infinite series

    Hi
    I'm not sure if it belongs in calculus but here it is anyway:

    Find the sum of \displaystyle\sum_{i=1}^{\infty}\frac{i^2}{a^i} for a constant a

    Wolframalpha yielded \frac{a(a+1)}{(a-1)^3} but I would like to know how to derive it by hand.
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  2. #2
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    Quote Originally Posted by oleholeh View Post
    Hi
    I'm not sure if it belongs in calculus but here it is anyway:

    Find the sum of \displaystyle\sum_{i=1}^{\infty}\frac{i^2}{a^i} for a constant a

    Wolframalpha yielded \frac{a(a+1)}{(a-1)^3} but I would like to know how to derive it by hand.
    The standard trick is to do it by differentiation, for consider this (assuming |a|<1, of course)

    \frac{d}{da}\sum\limits_{i=1}^\infty\left(\frac{1}  {a}\right)^i = \sum\limits_{i=1}^\infty\frac{d}{da}\left(\frac{1}  {a}\right)^i=\sum\limits_{i=1}^\infty i\left(\frac{1}{a}\right)^{i-1}=\frac{d}{da}\frac{1}{1-a}

    Now, apply this trick twice and you find that

    \sum\limits_{i=1}^\infty \frac{i^2}{a^i}=a\cdot\frac{d}{da}\left(a\cdot \frac{d}{da}\frac{1}{1-a}\right)
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