Value of Series $\sum_{n=1}^{\infty} \frac{1}{1-x^n}$

Asked Dec 19 '19 at 14:34

Active Dec 19 '19 at 14:47

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i have encountered the following series on a Physics problem: \begin{equation} S = \sum_{n=1}^{\infty} \frac{1}{1-x^n} \end{equation}

I would like to know if there is any specific Function assigned to this series. Thanks for you attention.

edited Dec 19 '19 at 14:45

asked Dec 19 '19 at 14:34

Júlio Oliveira

1

Does it not converge for $|x|>1$? – Fimpellizzeri Dec 19 '19 at 14:41
It seems that the series converges whenever $|x|>1$. W|A suggests
$$\sum_{n=1}^{\infty} \frac{1}{1-x^n} = \frac{\psi^{(0)}_{1/x} (1) + \log(x-1) + \log \frac{1}{x }}{\log x}$$

when $x>1$.
– Tolaso Dec 19 '19 at 14:42
Mathematica gives $$\frac{\psi _{\frac{1}{x}}^{(0)}(1)+\log (x-1)+\log \left(\frac{1}{x}\right)}{\log (x)}$$ where $\psi_q$ is the Polygamma function. The series converges when $|x|>1$ – Maximilian Janisch Dec 19 '19 at 14:42
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@Tolaso Well it seems we did the same thing – Maximilian Janisch Dec 19 '19 at 14:42
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@MaximilianJanisch Yep. Hehe! Great minds think alike. :) – Tolaso Dec 19 '19 at 14:43
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See also Lambert Series. – Fabio Lucchini Dec 19 '19 at 14:44
Thank you very much – Júlio Oliveira Dec 19 '19 at 14:46

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