# There's something strange about $\sum \frac 1 {\sin(n)}$.

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...

• 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. – Randall Jan 23 at 17:13
• Could you explain what you mean by "rational approximations for $\pi$ and whatnot"? – clathratus Jan 23 at 17:14
• @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. – Descartes Before the Horse Jan 23 at 17:18
• 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. – Steven Stadnicki Jan 23 at 21:41
• 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 $355$. – Daniel Fischer Jan 23 at 21:54