Conditions for the Following Series to Converge

Question

The problem is to show on which conditions the following series converge $$ \sum_{n=0}^\infty\left({\sum_{k=1}^{n}a^k}{(-1)^{(n-k)+1}\frac{(n-k)^2}{e^{n-k}}} \right) $$ I tried to simplify the series by writing their Cauchy product as $$ \sum_{n=0}^\infty\left({\sum_{k=1}^{n}a^k}{(-1)^{(n-k)+1}\frac{(n-k)^2}{e^{n-k}}} \right) \leq \left(\sum_{n=0}^\infty a^n \right) \left(\sum_{n=1}^\infty (-1)^{(n+1)}\frac{(n)^2}{e^{n}} \right) $$ However, first I need to prove both series converge absolutely. The series on the left-hand converge absolutely if $|a|<1$ , but I do not know on which conditions the second one also converge absolutely. So my question is to find the conditions in which the following series converge absolutely $$ \left(\sum_{n=1}^\infty (-1)^{(n+1)}\frac{(n)^2}{e^{n}} \right) $$