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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:

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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 $[360,700]$:

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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 $[700,1050]$:

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And I observe similar behavior on similar intervals.

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

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

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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...

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    $\begingroup$ I guess I wouldn't be shocked that reciprocating a periodic function which is often close to $0$ would do this. Maybe I'm not seeing it. $\endgroup$ – Randall Jan 23 at 17:13
  • $\begingroup$ Could you explain what you mean by "rational approximations for $\pi$ and whatnot"? $\endgroup$ – clathratus Jan 23 at 17:14
  • $\begingroup$ @clathratus integers $n$ can get arbitrarily close to some integer (and trivially, rational) multiples of $\pi$, and if some integer is close to an integer multiple of $\pi$, its corresponding term in our sum becomes arbitrarily large, and so the terms in our sum are not bounded and hence it does not converge. $\endgroup$ – Descartes Before the Horse Jan 23 at 17:18
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    $\begingroup$ I haven't pieced together the details yet, but I'm going to say about 75% that this has to do with rational approximations to $\pi$ and what amount to 'harmonics' in the sum where e.g. the fact that you have peaks every sixth $n$ in the early range is related to the fact that $2\pi\approx 6$ so that $1/\sin(n+6)\approx 1/\sin(n)$. Similarly, the wavefronts for $n\lt 1000$ are almost certainly spaced 44 units apart corresponding to $14\pi\approx 44$. I wouldn't be surprised if there's some very clever Fourier-esque transform that makes this explicit. $\endgroup$ – Steven Stadnicki Jan 23 at 21:41
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    $\begingroup$ You get a huge jump when $n$ is the numerator of a convergent of $\pi$. In between two such numerators, you get something close to "periodic" behaviour with smaller jumps from earlier convergents. It's a pity that between your second and third plots you skipped the two large jumps close together from $333$ and $355$. $\endgroup$ – Daniel Fischer Jan 23 at 21:54

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