I'm trying to apply de Cauchy criterion to the series $\sum_{n=0}^{\infty} 1/(zn)!$ with $z \in \mathbb{Z}^+$ in order to find an $N \in \mathbb{N}$ such that the finite evaluation of the series up to $N$ has a accuracy of 6 decimals. In order to find this $N$, let $N\leq n<m$:
$$\sum_{i=n}^m \frac{1}{(zi)!} \leq \sum_{i=n}^m \frac{1}{(zn)!}=\frac{m-n}{(zn)!}< \frac{m}{(zn)!} < \frac{m}{(zN)!}.$$
I want the last term to be smaller than $\varepsilon > 0$, but then $N$ will be depend of $m$. Could you find a way such that $m$ doesn't appear in the last term?