Stirling's approximation

James Stirling (1692-1770, Scotland)

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

Because of Euler-MacLaurin formula

$\sum _{k=1}^N\ln k=\underset{1}{\overset{N}{\int }}\ln xdx+\sum _{k=1}^p\frac{B_{2k}}{2k(2k-1)}(\frac{1}{n^{2k-1}}-1)+R$

where B₁ = −1/2, B₂ = 1/6, B₃ = 0, B₄ = −1/30, B₅ = 0, B₆ = 1/42, B₇ = 0, B₈ = −1/30, ... are the Bernoulli numbers, and R is an error term which is normally small for suitable values of p.

Then, for large N,

$\ln N!\sim \underset{1}{\overset{N}{\int }}\ln xdx\sim N\ln N-N$