Stirling's approximation

NOTE

$$\ln (n!)=n\ln n-n+O(\ln n)$$

explanation

$\ln (n!)-n\ln n+n\le c\ln n\exists c>0$ $\frac{n!}{\frac{n^n}{e^n}}\le n^c$ $n!\le {\left(\frac{n}{e}\right)}^n\cdot n^c$

Another approach $n!<{\left(\frac{n}{2}\right)}^n<n^n⟹{\lim }_{n\to \infty }\frac{n^n}{n!}=\infty$