Stirling’s Approximation

Approximating Factorials. Factorials are hard to do calculus with. Use Stirling’s approximation. \(N!\approx N\ln N-N\)

\(\ln N! = \ln 1 + \cdots + \ln N = \sum_{n=1}^N\ln n = \sum_{n=1}^N\int_{n-1/2}^{n+1/2}\ln x\:dx = \int_{1/2}^{N+1/2}\ln x\:dx = \int_0^N\ln x dx= N\ln N-N\)

Note, \(\ln n\approx\int_{n-\frac{1}{2}}^{n+\frac{1}{2}}\ln x\:dx\)

Stirling’s: \(N! = \sqrt{2\pi N}\left(\frac{N}{e}\right)^N\).