Prove or disprove the recursively defined sequence is convergent. The sequence $\{a_n\}$ is defined by $a_1=1, a_2=0$ and $a_{n+2}=a_{n+1}+\displaystyle\frac{a_n}{n^2}$ for $n\in \mathbb{N}$.
Since $\displaystyle\frac{1}{n^2}$ is summable, when $n$ is large, the sequence is something like $a_n=a_{n-1}+\displaystyle\sum_{i\leq n-2}\frac{a_i}{i^2}$, so I think the sequence should be convergent.
Then I want to use the Monotone convergent theorem, i.e. to show $\{a_n\}$ is monotonic and bounded.
For monotonic, it is easy to see that $\{a_n\}$ is increasing.
But for the upper bound, assuming $\{a_n\}$ converges and taking the limit $n\to \infty$ does not give any hints for me to find a suitable upper bound. I have also used computer programs to compute up to the 10000th term, but it seems that $\{a_n\}$ is still increasing, does not converges to a certain number.
So I wonder if it is convergent or not.

Hint: Show that $a_n \leq \prod_{i \leq n} (1+\frac  1{i^{3/2}})$ for all $n \geq 4$.  The infinite product $\prod_{i \leq n} (1+\frac  1{i^{3/2}})$ is convergent because $\sum_n \frac 1 {n^{3/2}} <\infty$.
[ Above inequality may hold for $n <4$ also but I found it easy to verify it for $n \geq 4$].
