The problem is to check whether $\sum_{n=1}^\infty a_n$ converges, where $a_1=1, a_{n}=\sin a_{n-1}$.
TLDR: I tried to apply Raabe's test:
$\lim_{n\to\infty} n\left(\left|\frac{a_n}{a_{n+1}}\right|-1\right)=R$, if $R<1$ then the series is divergent, if $R>1$ then it converges. I calculated values of this sequence for some big $n$ values using Python, and it seems that it converges to $R=\frac{1}{2}$, but I don't know how to prove that analytically. Please help.
Optional information:
This is not homework, this is me preparing for an admission exam using problems from previous years exams.
Ratio test is inconclusive, because $\lim_{x \to 0} \frac{\sin x}{x} = 1$. I don't know how to prove the limit used in root test, but a python program that calculates $a_n^{\frac{1}{n}}$ for very high $n$ is a great evidence that it is inconclusive as well.
The series can't be compared with a converging geometric series either.
