Stirling's approximation

James Stirling (1692-1770, Scotland)

$\left.\ln N!\right.=\ln 1+\ln 2+\ln 3+...+\ln N$

$~=\sum _{k=1}^{N}\ln k$

$~\approx \int _{1}^{N}\ln xdx$

$~=\left[x\ln x-x\right]_{1}^{N}$

$~=N\ln N-N+1$

Thus, for large N

$\ln N!\approx N\ln N-N$