I've been studying infinite series recently and believe I came across a counterintuitive (at least to me) result in the past from a textbook that I can't seem to find now. Is it possible to show $$\sum_{n=1}^\infty \frac{\csc(n)}{n}$$ converges? I actually think the result I saw was something along the lines of $\displaystyle\sum_{n=1}^\infty\frac{\csc(\sqrt{3}\, n)}{n}$ being a convergent series, though I don't remember the exact form and my internet searches haven't turned up anything that has been particularly helpful with this problem.
Plugging the series into WolframAlpha as is shows that it diverges, but computing various values for finite series shows that the value stabilizes around $-77.5$ (here is the sum from $n=1$ to $n=3000$) -- I computed various values up to $n=5000$
I did find related series (such as the Flint Hills Series and the Cookson Hills Series) and this potentially-related MSE question.
The series under consideration is the following result due to Hardy:
$$\sum_{n=1}^\infty\frac{1}{n^3\sin(n\pi\sqrt{2})}=-\frac{13\pi^3}{360\sqrt{2}}$$
This result can also be found in Titchmarsh's The Theory of Functions textbook.