Clearly, $$\sum_{n=1}^\infty \frac 1{\sin(n)}$$ Does not converge (rational approximations for $\pi$ and whatnot.) For fun, I plotted $$P(x)=\sum_{n=1}^x \frac 1{\sin(n)}$$ For $x$ on various intervals. At first, I saw what you might expect:

Which is $P(x)$ for $x \in [0,20]$ and then $[0,300]$. Seems a little self-similar, but whatever. Then I looked at $P(x)$ on the interval $[360,700]$:

OK, that looks suspiciously like $P(x)$ on the interval $[0,300]$, but I'll toss out this coincidence as 'probably has to do with $\pi$ being irrational.' Here is $P(x)$ on $[700,1050]$:

And I observe similar behavior on similar intervals.

**Putting it all together,** here is $P(x)$ on $[0,20000]$:

It's converging? Not quite. Here is $P(x)$ on $[20000,100000]$:

So again, we're seeing the function 'get closer and closer, then get farther and farther, all while alternating' from some value, just as we saw on the smaller intervals. I suspect that if my computer could handle $P(x)$ on $[100000,200000]$, we would see the same thing (on a larger scale), though I'm not sure.

So: what's going on here? How can we explain this fractal-ish behavior?

**Edit:** I wonder if $P:\mathbb{N} \to \mathbb{R}$ is injective...