Does a series converge or diverge?

Question

Let us consider the sequence $x_n$ defined by the recurence relation $x_1=1,\, x_{n+1}=\sin(x_n),\, n=1,2\dots,$ and a series $\sum_{n=1}^\infty x_n$. Numeric calculations suggest its divergence. Is it true?

Answer 1

It diverges by comparison test, since $x_n \sim \sqrt{\frac{3}{n}}$ as $n\to +\infty$. See, e.g., Convergence of $\sqrt{n}x_{n}$ where $x_{n+1} = \sin(x_{n})$