階乗(factorial)の対数(logarithm)の微分(differential calculus)

スターリングの近似(Stirling's approximation)

$$\begin{array}{rcl} \displaystyle n! & \simeq & \sqrt{2\pi n}\left(\dfrac{n}{e}\right)^n\\ \end{array}$$

階乗の対数

$$\begin{array}{rcl} \displaystyle \log(n!)&\simeq& \log\left( \sqrt{2\pi n}\left(\dfrac{n}{e}\right)^n \right)\\ &\simeq& \log\left( \sqrt{2\pi n}\,n^n\,e^{-n} \right)\\ \displaystyle &\simeq& \log\left( \sqrt{2\pi} \right) \displaystyle + \log\left( \sqrt{n} \right) \displaystyle + \log\left(n^n \right) \displaystyle + \log\left(e^{-n} \right)\\ \displaystyle &\simeq& \log\left( \sqrt{2\pi} \right) \displaystyle + \frac{1}{2}\log\left( n \right) \displaystyle + n\log\left(n \right) \displaystyle - n \log\left(e \right)\\ \displaystyle &\simeq& \log\left( \sqrt{2\pi} \right) \displaystyle + \frac{1}{2}\log\left( n \right) \displaystyle + n\log\left(n \right) \displaystyle - n\\ \end{array}$$

階乗の対数の微分

$$\begin{array}{rcl} \displaystyle \left(\log(n!)\right)' \displaystyle &\simeq& \left(\log\left( \sqrt{2\pi} \right)\right)' \displaystyle + \left(\frac{1}{2}\log\left( n \right)\right)' \displaystyle + \left(n\log\left(n \right)\right)' \displaystyle - \left(n\right)'\\ \displaystyle &\simeq& \left(\log\left( \sqrt{2\pi} \right)\right)' \displaystyle + \left(\frac{1}{2}\log\left( n \right)\right)' \displaystyle + \left(n(\log\left(n \right))'+(n)'\log\left(n \right)\right) \displaystyle - \left(n\right)'\\ \displaystyle &\simeq& 0 \displaystyle + \frac{1}{2}\left(\frac{1}{n}\right) \displaystyle + \left(n\left(\frac{1}{n}\right)+1\,\log\left(n \right)\right) \displaystyle - 1\\ \displaystyle &\simeq& \frac{1}{2n}+1+ \log\left(n \right)-1\\ \displaystyle &\simeq& \frac{1}{2n}+\log\left(n \right) \end{array}$$

\(n\)が十分に大きい場合

$$\begin{array}{rcl} \displaystyle \left(\log(n!)\right)' \displaystyle &\simeq& \frac{1}{2n}+\log\left(n \right)\\ \displaystyle &\simeq& 0+\log\left(n \right)\,\dotso\,nが十分い大きい \frac{1}{2n} \to 0\\ \displaystyle &\simeq& \log\left(n \right) \end{array}$$