Convergence and sum of a series How could I prove that the following series does converge? 
$$\sum_{k = 1}^{+\infty}\ \frac{e^{k!}}{k^{k!}}$$
And how to determine its total sum? 
I think that that series has to converge, because I took ratio test, $n$-th root test and so on.My problem is to compute the whole sum. Any idea? I tried with Stirling too but seems messy.

For convergence, use the root test:
$$
\sqrt[k]{  \frac{e^{k!}}{k^{k!}} }=\left( \frac{e}{k} \right) ^{(k-1)!} \to 0
$$
as for $k >7$ you have
$$
0< \left( \frac{e}{k} \right) ^{(k-1)!} < \left( \frac{1}{2} \right) ^{(k-1)!} < \left( \frac{1}{2} \right) ^{k-1}
$$
For the limit, it would surprise me if we can calculate it as a closed form, but maybe I am missing something.
