Let $\displaystyle \Gamma_N(s) = \frac{N^s(N-1)!}{s(s+1)\cdot \ldots \cdot (s+N-1)}$.

Show, for every $\displaystyle N \in \mathbb{N}$ that

$\displaystyle \int_0^N \displaystyle \left(1 - \frac{t}{N}\right)^N t^{s-1}dt = N^s \sum_{k=0}^N \frac{(-1)^k {N \choose k}}{k+s} = \frac{N}{N+s}\Gamma_N(s)$.

Hint:For the second inequality, divide both sides by $\displaystyle N^s$ and compare the resulting rational funtions in terms of their poles.