I'm unsure for what $x$ the following series converges/diverges:

$$\sum_{n=2}^{\infty} \frac{n}{x^n-1}$$

WolframAlpha is neither able to find an answer (with standard computing time, does anyone have WolframAlpha Pro?). I'm not that experienced with determining whether serieses converge or not but I tried to apply the ratio test:

\begin{align} L &= \lim _{n\to \infty}|{\frac {a_{n+1}}{a_{n}}}| \\\\ &= \lim _{n\to \infty}|{\frac {(n+1)x^n-n-1}{nx^{n+1}-n}}| \end{align}

...but I don't know how to continue from here.

So does this series converge or diverge (and for what $x$)? Any ideas?

  • $\begingroup$ What happens for different values of $x$ ? Are there any values you can immediately discount because of the limit of the series? $\endgroup$ – Ninad Munshi Apr 16 at 11:33

If $|x| <1$ then $\frac n {x^{n}-1}$ does not tend to $0$. Hence the series does not converge in this case. If $|x|>1$ then $|\frac n {x^{n}-1} |\leq \frac {2n} {|x|^{n}}$ for $n$ sufficiently large and $\sum \frac {2n} {|x|^{n}} <\infty$ by ratio test. I leave the case $x=1$ and $x=-1$ to you.

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