Convergence of series involving iterated $ \sin $

Solution 1:

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.