A cute approximation for $\cot(2\pi x)$(!?)
Here is another series expansion which results from exploiting the poles of $\cot(x)$,
$$ \cot(x)= \frac{1}{x} + \sum_{{k=-\infty}_{k\neq0}}^{\infty}\left( \frac{1}{x-k\pi}+ \frac{1}{k\pi} \right). $$
For more details on the method, see here starting from page $101$.