TITLE: Infinite series that surprisingly converge?
QUESTION [14 upvotes]: I couldn't find any substantial list of 'strange infinite convergent series' so I wanted to ask the MSE community for some. By strange, I mean infinite series/limits that converge when you would not expect them to and/or converge to something you would not expect.
My favorite converges to Khinchin's (sometimes Khintchine's) constant, $K$. For almost all $x \in 
\mathbb{R}$ (those for which this does not hold making up a measure zero subset) with infinite c.f. representation: 
$$x = a_0 + \frac{1}{a_1+\frac{1}{a_2+\frac1{\ddots}}}$$
We have:
$$\lim_{n \to \infty} =\root n \of{\prod_{i=1}^na_i} = \lim_{n \to \infty}\root n \of {a_1a_2\dots a_n} = K$$
Which is...wow! That it converges independent of $x$ really gets me.

REPLY [6 votes]: Let $x_n$ be the nth positive solution of $\csc(x)=x$, i.e. $x_1\approx 1.1141$, $x_2\approx 2.7726$, etc. Then,
$$\sum_{n=1}^{\infty}\frac{1}{x_n^2}=1$$

Edit: Even more surprisingly, if we define $s(k)=\sum x_n^{-k}$, then we have the generating function
\begin{align*}
\sum_{k=1}^{\infty}s(2k)x^{2k}
&=\frac{x}{2}\left(\frac{1+x\cot(x)}{\csc(x)-x}\right) \\
&=x^2+\frac{2x^4}{3}+\frac{21x^6}{40}+\frac{59x^8}{140}+\frac{24625x^{10}}{72576}+\cdots
\end{align*}
Unfortunately it seems that, as with the Riemann zeta function, the values of $s$ at odd integers are out of reach.