Equivalent of a special Generalized hypergeometric function

Question

Is there any equivalent or tight upper bound with an "elementary function" of following generalized hypergeometric function: ${}_k F_{k-1}(2,\dots,2,1-m;1,\dots,1;-1)$

when especially

1. $m$ and $k$ are fixed
2. $m$ tend to $+\infty$.
3. $m$ and $k=\lambda m$ tend to $+\infty$

I try to use DLMF and following answer:

Approximation of a Generalized Hypergeometric Function

but I can't seem to make it through.