# Thread: The sum of this infinite series?

1. ## 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.

2. ## 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?

3. ## Re: The sum of this infinite series?

closed-form, thanks. In terms hopefully of some special function (hypergeometric, Jacobi theta, ...)

4. ## 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?

5. ## 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 $n\to \infty$.

6. ## Re: The sum of this infinite series?

I was just wondering... I need it to compute a correlation for a bivariate distribution

7. ## Re: The sum of this infinite series?

Originally Posted by SlipEternal
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

8. ## Re: The sum of this infinite series?

Originally Posted by SlipEternal
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...