How do I evaluate the following finite sum over $k$


$1+\frac{1}{2^n}+\frac{1}{2^n3^n}+\frac{1}{2^n3^n4^n}+\cdots+\frac{1}{2^n3^n\cdots k^n}$


or if there is an expression of this sum in terms of other known numbers ??



How do I calculate this sum $\sum_k(k!)^{-n}$?
$${}\_1F\_n(1;2,2,\ldots 2; 1) - \frac{{}\_1F\_{n}(1;2+k,2+k,\ldots 2+k; 1)}{((k+1)!)^n} $$


