Is it possible to find the point of convergence of $\prod_1^\infty k^{\frac1{k!}}$
$K!=k(k-1)!$.
My attempt:
If $S_n=\prod_1^\infty k^{\frac1{k!}}$ then $\ln S_n=\sum_1^\infty \frac{\ln k}{k!}< \sum\frac1{k!}=e$
So $S_n$ is converges but I don't know if it is possible to find the point of convergence or not.
It's possible to rephrase your product as a nested radical, and sometimes these have closed-form solutions, like $\sqrt{2 \sqrt {2 \sqrt {2 \ldots}}} = 2$. However no one even knows a closed-form solution for the seemingly almost as innocent $\sqrt{2 \sqrt{3 \sqrt{4 \ldots}}}$. There all the nested radicals are square-roots. Your nested radicals are $k$th roots for increasing $k$. Thus it seems highly unlikely that anyone will be able to come up with a closed form for your expression, at least, not without some monumental effort.