Numerical calculations and some theory leads to the suggestion that
$$\cot(2\pi x) \rightarrow\frac{1}{2\pi}\sum_r \frac{1}{x-r}$$
where $r$ ranges over all the roots of $B_{2n+1}$ (Bernoulli polynomial) as $n\rightarrow \infty$ and $n \in \mathbb{N}$.
Does this converge to $\cot(2\pi x)$? If so, how fast? Do you have pointers to books, articles?
Here is one article that is relevant for a start:
Uniform Convergence Behavior of the Bernoulli Polynomials
The theory behind it is really just Corollary 2.1, page 3 from that article and that for
$P(x)$ and $Q(x) = (x-\alpha_1)(x-\alpha_2) \cdots (x-\alpha_n)$ polynomials $\textrm{deg }P < \textrm{deg }Q$, $\alpha_i$ distinct, then
$$\frac{P(x)}{Q(x)} = \sum_{i=1}^n \frac{P(\alpha_i)}{Q'(\alpha_i)}\frac{1}{(x-\alpha_i)}$$ partial fractions Wikipedia (also $B_n'(x)=nB_{n-1}(x))$
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 $101$.