when is this series convergent.

$$ \sum_{\ell=0}^n \left( \sum_{k=0}^n \frac{k^2 x^k y^\ell}{\ell!} \right) = \sum_{\ell=0}^n \left( \frac{y^\ell}{\ell!} \sum_{k=0}^n k^2 x^k \right) $$ The step above can be done because $\dfrac{y^\ell}{\ell!}$ does not change as $k$ goes from $0$ to $n$.

This next step can be done because $\displaystyle\sum_{k=0}^n k^2 x^k$ does not change as $\ell$ goes from $0$ to $n$: $$ \sum_{\ell=0}^n \left( \frac{y^\ell}{\ell!} \left( \sum_{k=0}^n k^2 x^k \right) \right) = \left( \sum_{\ell=0}^n \frac{y^\ell}{\ell!} \right) \left( \sum_{k=0}^n k^2 x^k \right) $$ Now you just need to consider them separately. A ratio test does the first one instantly if you know things like $52!/53! = 1/53$. A ratio test also handles the second one if you know how to find things like $\displaystyle \lim_{k\to\infty} \dfrac{(k+1)^2}{k^2}$.