Stirling's approximation

James Stirling (1692-1770, Scotland)

$\ln N!=\ln 1+\ln 2+\ln 3+...+\ln N=\sum _{k=1}^N\ln k$

$\approx \underset{1}{\overset{N}{\int }}\ln xdx$

$={[x\ln x-x]}_1^N$

$=N\ln N-N+1$

Thus, for large N

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