階乗(factorial)の逆数(reciprocal)の和(無限級数(infinite series))

$$\begin{array}{rcl} \displaystyle f(x) &=& \displaystyle \sum_{k=0}^{\infty}\frac{f^{(k)}(a)}{x!}(x-a)^k\,\dotso\,a点まわりのテイラー展開\\ &=& \displaystyle \frac{1}{0!}f^{(0)}(a)(x-a)^0+\frac{1}{1!}f^{(1)}(a)(x-a)^1+\frac{1}{2!}f^{(2)}(a)(x-a)^2+\dotsb\\ \end{array}$$

\(e^x\)のマクローリン展開

$$\begin{array}{rcl} \displaystyle \mathrm{e}^x　 &=& \displaystyle \sum_{k=0}^{\infty}\frac{1}{k!}\left\{\left(\frac{\mathrm{d}^k}{\mathrm{d}x^k}\mathrm{e}^x\right)|_{x=0}\right\}(x-0)^k \,\dotso\,0点まわりのテイラー展開(Taylor\,series)=マクローリン展開(Maclaurin\,expansion)\\ &=& \displaystyle \frac{1}{0!}\left\{\left(\frac{\mathrm{d}^0}{\mathrm{d}x^0}\mathrm{e}^x\right)|_{x=0}\right\}(x-0)^0 \displaystyle +\frac{1}{1!}\left\{\left(\frac{\mathrm{d}^1}{\mathrm{d}x^1}\mathrm{e}^x\right)|_{x=0}\right\}(x-0)^1 \displaystyle +\frac{1}{2!}\left\{\left(\frac{\mathrm{d}^2}{\mathrm{d}x^2}\mathrm{e}^x\right)|_{x=0}\right\}(x-0)^2 \displaystyle +\dotsb\\ &=& \displaystyle \frac{1}{0!}\left\{\left(\mathrm{e}^x\right)|_{x=0}\right\}(x-0)^0 \displaystyle +\frac{1}{1!}\left\{\left(\mathrm{e}^x\right)|_{x=0}\right\}(x-0)^1 \displaystyle +\frac{1}{2!}\left\{\left(\mathrm{e}^x\right)|_{x=0}\right\}(x-0)^2 \displaystyle +\dotsb\\ &=& \displaystyle \frac{1}{0!}\,1\,x^0 \displaystyle +\frac{1}{1!}\,1\,x^1 \displaystyle +\frac{1}{2!}\,1\,x^2 \displaystyle +\dotsb \,\dotso\,a^0=1\\ &=& \displaystyle \sum_{k=0}^{\infty}\frac{1}{k!}x^k\\ &=& \displaystyle \sum_{k=0}^{\infty}\frac{x^k}{k!}\\ \end{array}$$ $$\begin{array}{rcl} \displaystyle \sum_{k=0}^{\infty}\frac{x^k}{k!}&=&\displaystyle \mathrm{e}^x\\ \displaystyle \sum_{k=0}^{\infty}\frac{1}{k!}=\sum_{k=0}^{\infty}\frac{1^k}{k!}&=&\displaystyle \mathrm{e}^1=\mathrm{e}\,\dotso\,x=1\\ \end{array}$$

\(e^{Cx}\)のマクローリン展開

$$\begin{array}{rcl} \displaystyle \mathrm{e}^{Cx}　 &=& \displaystyle \sum_{k=0}^{\infty}\frac{1}{k!}\left\{\left(\frac{\mathrm{d}^k}{\mathrm{d}x^k}\mathrm{e}^{Cx}\right)|_{x=0}\right\}(x-0)^k \,\dotso\,0点まわりのテイラー展開(Taylor\,series)=マクローリン展開(Maclaurin\,expansion)\\ &=& \displaystyle \frac{1}{0!}\left\{\left(\frac{\mathrm{d}^0}{\mathrm{d}x^0}\mathrm{e}^{Cx}\right)|_{x=0}\right\}(x-0)^0 \displaystyle +\frac{1}{1!}\left\{\left(\frac{\mathrm{d}^1}{\mathrm{d}x^1}\mathrm{e}^{Cx}\right)|_{x=0}\right\}(x-0)^1 \displaystyle +\frac{1}{2!}\left\{\left(\frac{\mathrm{d}^2}{\mathrm{d}x^2}\mathrm{e}^{Cx}\right)|_{x=0}\right\}(x-0)^2 \displaystyle +\dotsb\\ &=& \displaystyle \frac{1}{0!}\left\{\left(\mathrm{e}^{Cx}\right)|_{x=0}\right\}(x-0)^0 \displaystyle +\frac{1}{1!}\left\{\left(C\mathrm{e}^{Cx}\right)|_{x=0}\right\}(x-0)^1 \displaystyle +\frac{1}{2!}\left\{\left(C^2\mathrm{e}^{Cx}\right)|_{x=0}\right\}(x-0)^2 \displaystyle +\dotsb\\ &=& \displaystyle \frac{1}{0!}\,1\,x^0 \displaystyle +\frac{1}{1!}\,C\,x^1 \displaystyle +\frac{1}{2!}\,C^2\,x^2 \displaystyle +\dotsb \,\dotso\,a^0=1\\ &=& \displaystyle \frac{1}{0!}\,C^0\,x^0 \displaystyle +\frac{1}{1!}\,C^1\,x^1 \displaystyle +\frac{1}{2!}\,C^2\,x^2 \displaystyle +\dotsb \,\dotso\,a^0=1\\ &=& \displaystyle \sum_{k=0}^{\infty}\frac{(Cx)^k}{k!}\\ \end{array}$$ $$\begin{array}{rcl} \displaystyle \sum_{k=0}^{\infty}\frac{(Cx)^k}{k!} &=& \mathrm{e}^{Cx}\\ \end{array}$$