Help in simplifying this nasty expression obtained after binomial expnasion

Question

I have arrived to the following expression and was wondering if anyone can help me further simplify to something nicer,

$$F= 1- [1-\text{exp} (- \alpha(N) ) ]^N= 1- \sum_{k=0}^{N} \binom{N}{k} \bigg(- \text{exp} (- \alpha(N) )\bigg)^k$$

where $$ \alpha(N)= (N!)^{-{1}/{N}}$$

Any ideas?

Answer 1

i have got $2-\left(1-e^{(N!)^{-1/N}}\right)^N$ after an ugly calculation.