Convergence or divergence of $a_{n+1}=a_n +\frac{a_{n-1}}{(n+1)^2}$

If $a_1=a_2=1$ and $a_{n+1}=a_n +\frac{a_{n-1}}{(n+1)^2}$ How to prove convergence of the sequence and its limit or divergence?

It is easy to see that the sequence is always positive and by that one can conclude it is strictly increasing sequence but proving an upper bound doesn't seem to be easy, so maybe it diverge?

I wrote a simple python code and found that all $a_n =1.4552363391830$ where $n = 200,000,000$ this might suggest that the sequence is bounded but it is worth noting that this sequence might have a very slow convergence or a very slow divergence because $a_n=1.4552\color{red}{291356494609}$ where $n=200,000$ so the sequence is able to gain about $10^{-5}$ from $2\times 10^5$ to $2\times 10^8$.

import mathx=1y=1for i in range(3, 1_000_000_000):    x=y+x/((2*i-3)**2)    y=x+y/((2*i-2)**2)    print(x)print(y)

$$a_{2*10^9} =1.4552364010671075$$

If this sequence diverges then how many terms one need for $a_n >x$ for any $x\in \mathbb{R}^+$? If this sequence converges can a closed form for a the limit exist ?