The sequence $a_n$ is defined as: $$a_1 = 1, a_2 = 2, a_{n+1} = 1 + a_1a_2...a_{n-1}+\left(a_1a_2...a_{n-1} \right)^2 \text{ for all } n\ge 2$$ Prove that the sequence $s_n = \sum_{k=1}^n{1/a_k} $ is convergent and find $\lim s_n, n \rightarrow \infty$.
What I have done: I constructed two sequences ${\{b_1=1, b_n=n(n-1), n\ge2\} }$ and ${\{c_n=2^{n-1}, n \ge 1\}}$ such that $b_n \le a_n \le c_n$ and both $\sum_{k=1}^n{1/b_k}$ and $\sum_{k=1}^n{1/c_k} $ are convergent with limit 2. I'm interested to know if this is correct and if there are other solutions.