Stirling’s Approximation

Approximating Factorials. Factorials are hard to do calculus with. Use Stirling’s approximation. $N!\approx N\ln N-N$

$\ln N!=\ln 1+\cdots +\ln N=\sum _{n=1}^N\ln n=\sum _{n=1}^N{\int }_{n-1/2}^{n+1/2}\ln xdx={\int }_{1/2}^{N+1/2}\ln xdx={\int }_0^N\ln xdx=N\ln N-N$

Note, $\ln n\approx {\int }_{n-\frac{1}{2}}^{n+\frac{1}{2}}\ln xdx$

Stirling’s: $N!=\sqrt{2\pi N}{\left(\frac{N}{e}\right)}^N$.