Stirling's Approximation

#Math #Physics
From Wikipedia

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

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

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

• Stirling's Approximation
• Better approximation by second

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

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