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Thread: The sum of this infinite series?

  1. #1
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    The sum of this infinite series?

    ... how can this infinite sum can be computed: $\sum_{k=1}^{\infty} a^k/(1-bc^k)$, with $0<a,b,c<1$?
    Thanks.
    Last edited by skeeter; May 16th 2017 at 08:02 AM. Reason: post moved to new thread
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  2. #2
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    Re: The sum of this infinite series?

    What are you looking for? Are you looking for radius of convergence? Or do you want a closed form for the sum?
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  3. #3
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    Re: The sum of this infinite series?

    closed-form, thanks. In terms hopefully of some special function (hypergeometric, Jacobi theta, ...)
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  4. #4
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    Re: The sum of this infinite series?

    Have you done any work on this?

    Edit: What makes you think there might be a closed form for this?
    Last edited by SlipEternal; May 16th 2017 at 09:33 AM.
    Thanks from skeeter
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  5. #5
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    Re: The sum of this infinite series?

    https://www.wolframalpha.com/input/?...infinity%7D%5D

    Here is what Wolframalpha comes up with when I put $a=0.5, b=0.125, c=0.25$. It is using the q-digamma function (which I have no idea what that even is). This is only giving you partial sums, so you would need to calculate the limit function as $n\to \infty$.
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  6. #6
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    Re: The sum of this infinite series?

    I was just wondering... I need it to compute a correlation for a bivariate distribution
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  7. #7
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    Re: The sum of this infinite series?

    Quote Originally Posted by SlipEternal View Post
    Have you done any work on this?

    Edit: What makes you think there might be a closed form for this?
    I was just wondering... I need it to compute a correlation for a bivariate distribution
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  8. #8
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    Re: The sum of this infinite series?

    Quote Originally Posted by SlipEternal View Post
    https://www.wolframalpha.com/input/?...infinity%7D%5D

    Here is what Wolframalpha comes up with when I put $a=0.5, b=0.125, c=0.25$. It is using the q-digamma function (which I have no idea what that even is). This is only giving you partial sums, so you would need to calculate the limit function as $n\to \infty$.
    Thanks! Indeed it would be nice if Mathematica provided a general formula for given a, b, c... but this would be a hard task, given they are bounded to the unit interval...
    Last edited by alcliver; May 17th 2017 at 12:03 AM.
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