式展開: 階乗(factorial)の対数(logarithm)の微分(differential calculus)

\begin{array}{rcl} n! & ≃ & \sqrt{2 π n} {(\frac{n}{e})}^{n} \end{array}

\begin{array}{rcl} \log (n!) & ≃ & \log (\sqrt{2 π n} {(\frac{n}{e})}^{n}) \\ ≃ & \log (\sqrt{2 π n} n^{n} e^{- n}) \\ ≃ & \log (\sqrt{2 π}) + \log (\sqrt{n}) + \log (n^{n}) + \log (e^{- n}) \\ ≃ & \log (\sqrt{2 π}) + \frac{1}{2} \log (n) + n \log (n) - n \log (e) \\ ≃ & \log (\sqrt{2 π}) + \frac{1}{2} \log (n) + n \log (n) - n \end{array}

\begin{array}{rcl} {(\log (n!))}^{'} & ≃ & {(\log (\sqrt{2 π}))}^{'} + {(\frac{1}{2} \log (n))}^{'} + {(n \log (n))}^{'} - {(n)}^{'} \\ ≃ & {(\log (\sqrt{2 π}))}^{'} + {(\frac{1}{2} \log (n))}^{'} + (n (\log (n))^{'} + (n)^{'} \log (n)) - {(n)}^{'} \\ ≃ & 0 + \frac{1}{2} (\frac{1}{n}) + (n (\frac{1}{n}) + 1 \log (n)) - 1 \\ ≃ & \frac{1}{2 n} + 1 + \log (n) - 1 \\ ≃ & \frac{1}{2 n} + \log (n) \end{array}

\begin{array}{rcl} {(\log (n!))}^{'} & ≃ & \frac{1}{2 n} + \log (n) \\ ≃ & 0 + \log (n) \dots n が 十 分 い 大 き い \frac{1}{2 n} \to 0 \\ ≃ & \log (n) \end{array}

式展開