Stirling's Approximation

From Wikipedia

$n!\approx n^n{e}^{-n}$

• Worse approximation
• $\ln (n!)=n\ln n-n$

$n!\approx \sqrt{2\pi n}{\left(\frac{n}{e}\right)}^n$

• Stirling’s Approximation
• Better approximation by second

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

• $O(\ln n)$ are the correction terms
• Next term is $\frac{1}{2}\ln (2\pi n)$