grandes-ecoles 2020 Q6

grandes-ecoles · France · centrale-maths1__psi Number Theory Divisibility and Divisor Analysis

Let $n \in \mathbb{N}^*$. Show that for all $k \in \mathbb{N}^*$, $$\left(\frac{n}{n+k}\right)^k \leqslant \frac{n! n^k}{(n+k)!} \leqslant 1.$$

Let $n \in \mathbb{N}^*$. Show that for all $k \in \mathbb{N}^*$,
$$\left(\frac{n}{n+k}\right)^k \leqslant \frac{n! n^k}{(n+k)!} \leqslant 1.$$