Does the series $\sum \sin^{(n)}(1)$ converge, where $\sin^{(n)}$ denotes the $n$-fold composition of $\sin$?

I'm trying to solve the following task

Sequence $\{a_n\}$ is given by the rule: $a_1 = 1,\: a_{n+1} = \sin (a_n)$. Does the series $\sum a_n$ converge?

Can you give me any hints how to solve it, cause i got totally stuck at the very beginning, please?

Solution 1:

The series diverges. To see this, first note that $$ a_1 = 1\ge 1 $$ and that, if $a_n \ge 1/n$, then $$ a_{n+1} = \sin(a_n) \ge \sin(1/n) > 1/(n+1) $$ By induction, we have $a_n \ge 1/n$ for all $n$. Since $\sum\frac{1}{n}$ diverges, so does $\sum a_n$.

Note that $(n+1)\sin(1/n) > 1$ can be shown by Taylor expansion.

Solution 2:

Just an intuition:

Since $sin(x)$ almost equals $x$ for values of $x$ near $0$, so it's like adding the same term infinitely with hardly any change in it...so it would diverge for sure!