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$\displaystyle\sum_{n=1}^\infty\frac{\cot(\pi nx)}{n^2+x^2}$

Is there a closed form for the series $\displaystyle\sum_{n=1}^\infty\frac{\cot(\pi nx)}{n^2+x^2}$?

A:

Expanding on @Smeagol’s suggestion, we have
$$
\frac{\cot(\pi nx)}{n^2+x^2} = \frac1{2n} \cdot \frac1{\frac{\pi nx}{n} + \pi x} = \frac1{2n} \cdot \frac1{1+\frac{\pi^2 n^2 x^2}{n^2 + x^2}}.
$$
Expanding the factor $\frac1{1+\frac{\pi^2 n^2 x^2}{n^2 + x^2}}$ using the Euler-Maclaurin formula, it is elementary to prove that
$$
\sum_{n=1}^N \frac1{n^\alpha} = \frac{N^{1-\alpha}}{\alpha – 1} + O(1).
$$
Therefore, we can write
\begin{align*}
\sum_{n=1}^\infty \frac{\cot(\pi nx)}{n^2+x^2}
&= \sum_{n=1}^\infty \frac1{2n} \cdot \frac1{1+\frac{\pi^2 n^2 x^2}{n^2 + x^2}} \\
&= \frac1{2x} + \sum_{n=1}^\infty \frac{(-1)^{n-

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A:

You can use WinRAR’s own arcadia utility to burn your image to disk, which is pretty simple in its instructions.
If you don’t have a broadband connection or don’t want to pay for a paid for service, I believe all you need to do is download the.exe utility, install the.bin driver(s), and then run the.exe.

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