Let $n$ be a non-zero natural integer and let $p$ be a prime number. Justify the formula $\nu_p(n!) = \sum_{k=1}^{+\infty} E\left(\frac{n}{p^k}\right)$ and show that $$\frac{n}{p} - 1 < \nu_p(n!) \leqslant \frac{n}{p} + \frac{n}{p(p-1)}$$
Let $n$ be a non-zero natural integer and let $p$ be a prime number. Justify the formula $\nu_p(n!) = \sum_{k=1}^{+\infty} E\left(\frac{n}{p^k}\right)$ and show that
$$\frac{n}{p} - 1 < \nu_p(n!) \leqslant \frac{n}{p} + \frac{n}{p(p-1)}$$