Sequences $(a_n), (b_n)$ are defined as follows: \begin{align*} a_0 = 0;\quad a_{n+1} = a_n + \sqrt{a_n^2+1}; \quad b_n = \frac{a_n}{2^n}; \quad n \geq 0. \end{align*} It is not difficult to prove that sequence $(b_n)$ has a limit when $n\to\infty$. Numerical simulation shows that this limit might be $\frac{2}{\pi}$. But I can't prove this.
I would like to get any insight which will help to settle this.
