<s>[INST] Convergence of series with $a_{n + 1} = \sin{a_n}$ Given the sequence  $a_n \mspace{10mu},\mspace{10mu}a_{n+1}=\sin(a_n)\mspace{10mu} and \mspace{10mu} a_1=1. $
Find if the series $\sum_{k=1}^{\infty}a_k$ converges or diverges.

First I found that $a_n$ is monotone decreasing sequence. If $a_n\in[0;\pi/2]$ then   $\sin(a_n)\leqslant a_n$. And $a_n\leq 0$.
$\lim_{n\rightarrow \infty}a_n=0.$
Ratio test gives $\lim_{n\rightarrow \infty}\frac{a_{n+1}}{a_n}=1$.
More investigation of the series is required.
 [/INST] As you mentioned, $(a_n)$ decreases to $0$. Therefore, you have
$$a_{n+1}^{-2} - a_n^{-2} = \sin(a_n)^{-2} - a_n^{-2} = \left(a_n - \frac{a_n^3}{6} + o(a_n^4)\right)^{-2} - a_n^{-2}$$ $$=a_n^{-2} \left[\left(1 - \frac{a_n^2}{6} + o(a_n^3)\right)^{-2} - 1 \right] = a_n^{-2} \left(1 +  \frac{a_n^2}{3} + o(a_n^2) - 1 \right) = \frac{1}{3} + o(1)$$
So the sequence $a_{n+1}^{-2} - a_n^{-2}$ converges to $\frac{1}{3}$. You can now use Cesaro to see that
$$a_n^{-2} \sim \frac{n}{3}$$
Therefore
$$a_n \sim \sqrt{\frac{3}{n}}$$
so by comparison, the series diverges.
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