Answer
The Ratio Test; converges.
Work Step by Step
We apply the Ratio Test. Write ${a_k} = \dfrac{{k!}}{{{{\rm{e}}^{{k^2}}}}}$.
Evaluate:
$\rho = \mathop {\lim }\limits_{k \to \infty } \dfrac{{{a_{k + 1}}}}{{{a_k}}} = \dfrac{{\left( {k + 1} \right)!}}{{{{\rm{e}}^{{{\left( {k + 1} \right)}^2}}}}}\cdot\dfrac{{{{\rm{e}}^{{k^2}}}}}{{k!}} = \mathop {\lim }\limits_{k \to \infty } \dfrac{{k + 1}}{{{{\rm{e}}^{2k + 1}}}} = 0$
Since $\rho = 0 \lt 1$, by the Ratio Test (Theorem 9.5.5), the series $\mathop \sum \limits_{k = 1}^\infty \dfrac{{k!}}{{{{\rm{e}}^{{k^2}}}}}$ converges.
