analysis - Convergence of series involving iterated $sin$

Hi all,

I've been trying to show the convergence or divergence of

$$\sum_{n=1}^\infty \frac{\sin^n 1}{n} = \frac{\sin 1}{1} + \frac{\sin \sin 1}{2} + \frac{\sin \sin \sin 1}{3} + \ \cdots$$

where the superscript means iteration (not multiplication, so it's not simply less than a geometric series -- I couldn't find the standard notation for this).

Problem is,

• $\sin^n 1 \to 0$ as $n \to \infty$ (which I eventually proved by assuming a positive limit and having $\sin^n 1$ fall below it, after getting its existence) helps the series to converge,

  
  
  
  

  but at the same time




• $\sin^{n+1} 1 = \sin \sin^n 1 \approx \sin^n 1$ for large $n$ makes it resemble the divergent harmonic series.

I would appreciate it if someone knows a helpful convergence test or a proof (or any kind of advice, for that matter).

In case it's useful, here are some things I've tried:

• Show $\sin^n 1 = O(n^{-\epsilon})$ and use the p-series. I'm not sure that's even true.


• Computer tests and looking at partial sums. Unfortunately, $\sum 1/n$ diverges very slowly, which is hard to distinguish from convergence.


• Somehow work in the related series
  $$\sum_{n=1}^\infty \frac{\cos^n 1}{n} = \frac{\cos 1}{1} + \frac{\cos \cos 1}{2} + \frac{\cos \cos \cos 1}{3} + \ \cdots$$
  which I know diverges since the numerators approach a fixed point.

Answer

A Google search has turned up an analysis of the asymptotic behavior of the iterates of $\sin$ on page 157 of de Bruijn's Asymptotic methods in analysis. Namely,

$$\sin^n(1)=\frac{\sqrt{3}}{\sqrt{n}}\left(1+O\left(\frac{\log(n)}{n}\right)\right),$$

which implies that your series converges.

Edit: Aryabhata has pointed out in a comment that the problem of showing that $\sqrt{n}\sin^n(1)$ converges to $\sqrt{3}$ already appeared in the question Convergence of $\sqrt{n}x_{n}$ where $x_{n+1} = \sin(x_{n})$ (asked by Aryabhata in August). I had missed or forgot about it. David Speyer gave a great self contained answer, and he also referenced de Bruijn's book. De Bruijn gives a reference to a 1945 work of Pólya and Szegő for this result.