Does the series $\sum \sin{(\sin{(...\sin{(1)}...)})}$ converge?

Question

Let $a_{n+1} = \sin{a_n}$ and $a_0 = 1$. Does the series $\displaystyle \sum_{i=0}^\infty a_i$ converge?

Here is my solution.

Let's prove by induction, that $a_n > \frac{1}{n}$.

We see that $\sin{(1)} > \frac{1}{2}$.

Suppose, that $a_n > \frac{1}{n}$. Then:

$$\underbrace{\sin{\sin{(...\sin{(1)}...)}} )}_\text{$n+1$ sines}\ ?\ \frac{1}{n+1}$$

$$\underbrace{\sin{\sin{(...\sin{(1)}...)}} )}_\text{$n$ sines}\ ?\ \mbox{arcsin}(\frac{1}{n+1})$$

If $\mbox{arcsin}(\frac{1}{n+1}) < \frac{1}{n}$ (becuase left part is bigger then it by induction hypothesis) then we are done. And it really is, because $\sin{\frac{1}{n}} > \frac{1}{n+1} \Leftrightarrow \frac{1}{n} - \frac{1}{n^33!} > \frac{1}{n+1}$ (here I used Teylor expansion for sine).

As $a_n > \frac{1}{n}$ our series is divergent by comparison test.

Is my reasoning correct?

Answer 1

Given $a_0=1$ and $a_{n+1}=\sin(a_n)$, it is pretty clear that $a_n\to 0$ as $n\to +\infty$.
On the other hand we may show that $a_n$ behaves like $\frac{C}{\sqrt{n}}$ for large values of $n$, hence the series $\sum_{n\geq 1}a_n$ is divergent by asymptotic comparison and the $p$-test. In a neighbourhood of the origin we have $\sin(x)=x-\frac{x^3}{6}+o(x^4)$, hence $a_{n+1}-a_n\sim -\frac{a_n^3}{6}$. The positive solutions of the differential equation $f'(x)=-\frac{f(x)^3}{6}$ have the form $f(x)=\frac{\sqrt{3}}{\sqrt{x+K}}$, hence $$ \lim_{n\to +\infty} \sqrt{n} a_n = \sqrt{3} \tag{1}$$ and $\sum_{n=1}^{N}a_n$ diverges like $\sum_{n=1}^{N}\frac{\sqrt{3}}{\sqrt{n}} \approx 2\sqrt{3N}$. The previous limit is indeed pretty well-known and it can be proved through the Stolz-Cesàro theorem too.