convergence divergence - Closed form of :$S n,m= sum_{k=1}^n (-1)^kbinom{n}{k}k^{-m!}$

I don't succed to get a closed form of the bellow sum using standard Binomial law , in order to know if this sum could be converge or not for $n\to +\infty$ ,is there any simple way or any algorithm to eavaluate the bellow sum :

$$S n,m= \sum_{k=1}^n (-1)^k\binom{n}{k}k^{-m!}$$

?

Answer

The convergence of $S_{n,m}$ can easily be determined when applying the so-called Dilcher's fromula

$$\sum_{1\le n_1\le\cdots\le n_M\le n}\;\prod_{j=1}^{M}\frac{1}{n_j} =\sum_{k=1}^{n}\binom{n}{k}\cdot\frac{(-1)^{k-1}}{k^M},$$

where $M,n\in\mathbb N$ (for more detail, see http://mathworld.wolfram.com/DilchersFormula.html).

Finally, set $M=m!$, $m\in\mathbb N$, to obtain

$$-S_{n,m} =\sum_{1\le n_1\le\cdots\le n_{m!}\le n}\;\prod_{j=1}^{m!}\frac{1}{n_j} \ge\sum_{k=1}^{n}\frac{1}{k}.$$

Consequently, one deduces that $S_{n,m}\to -\infty$ as $n\to\infty$.