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

$$\begin{array}{rcl}{\displaystyle f(x)}& =& {\displaystyle \sum _{k=0}^{\mathrm{\infty }}\frac{f^{(k)}(a)}{x!}(x-a{)}^k\dots a\mathrm{点}\mathrm{ま}\mathrm{わ}\mathrm{り}\mathrm{の}\mathrm{テ}\mathrm{イ}\mathrm{ラ}\mathrm{ー}\mathrm{展}\mathrm{開}}\\ & =& {\displaystyle \frac{1}{0!}{f}^{(0)}(a)(x-a{)}^0+\frac{1}{1!}{f}^{(1)}(a)(x-a{)}^1+\frac{1}{2!}{f}^{(2)}(a)(x-a{)}^2+\cdots }\end{array}$$

$e^x$のマクローリン展開

$$\begin{array}{rcl}{\displaystyle {\mathrm{e}}^x }& =& {\displaystyle \sum _{k=0}^{\mathrm{\infty }}\frac{1}{k!}\left\{\left(\frac{{\mathrm{d}}^k}{\mathrm{d}{x}^k}{\mathrm{e}}^x\right){|}_{x=0}\right\}(x-0{)}^k\dots 0\mathrm{点}\mathrm{ま}\mathrm{わ}\mathrm{り}\mathrm{の}\mathrm{テ}\mathrm{イ}\mathrm{ラ}\mathrm{ー}\mathrm{展}\mathrm{開}(Taylorseries)=\mathrm{マ}\mathrm{ク}\mathrm{ロ}\mathrm{ー}\mathrm{リ}\mathrm{ン}\mathrm{展}\mathrm{開}(Maclaurinexpansion)}\\ & =& {\displaystyle \frac{1}{0!}\left\{\left(\frac{{\mathrm{d}}^0}{\mathrm{d}{x}^0}{\mathrm{e}}^x\right){|}_{x=0}\right\}(x-0{)}^0{\displaystyle +\frac{1}{1!}\left\{\left(\frac{{\mathrm{d}}^1}{\mathrm{d}{x}^1}{\mathrm{e}}^x\right){|}_{x=0}\right\}(x-0{)}^1{\displaystyle +\frac{1}{2!}\left\{\left(\frac{{\mathrm{d}}^2}{\mathrm{d}{x}^2}{\mathrm{e}}^x\right){|}_{x=0}\right\}(x-0{)}^2{\displaystyle +\cdots }}}}\\ & =& {\displaystyle \frac{1}{0!}\left\{\left({\mathrm{e}}^x\right){|}_{x=0}\right\}(x-0{)}^0{\displaystyle +\frac{1}{1!}\left\{\left({\mathrm{e}}^x\right){|}_{x=0}\right\}(x-0{)}^1{\displaystyle +\frac{1}{2!}\left\{\left({\mathrm{e}}^x\right){|}_{x=0}\right\}(x-0{)}^2{\displaystyle +\cdots }}}}\\ & =& {\displaystyle \frac{1}{0!}1x^0{\displaystyle +\frac{1}{1!}1x^1{\displaystyle +\frac{1}{2!}1x^2{\displaystyle +\cdots \dots a^0=1}}}}\\ & =& {\displaystyle \sum _{k=0}^{\mathrm{\infty }}\frac{1}{k!}{x}^k}\\ & =& {\displaystyle \sum _{k=0}^{\mathrm{\infty }}\frac{x^k}{k!}}\end{array}$$ $$\begin{array}{rcl}{\displaystyle \sum _{k=0}^{\mathrm{\infty }}\frac{x^k}{k!}}& =& {\displaystyle {\mathrm{e}}^x}\\ {\displaystyle \sum _{k=0}^{\mathrm{\infty }}\frac{1}{k!}=\sum _{k=0}^{\mathrm{\infty }}\frac{1^k}{k!}}& =& {\displaystyle {\mathrm{e}}^1=\mathrm{e}\dots x=1}\end{array}$$

$e^{Cx}$のマクローリン展開

$$\begin{array}{rcl}{\displaystyle {\mathrm{e}}^{Cx} }& =& {\displaystyle \sum _{k=0}^{\mathrm{\infty }}\frac{1}{k!}\left\{\left(\frac{{\mathrm{d}}^k}{\mathrm{d}{x}^k}{\mathrm{e}}^{Cx}\right){|}_{x=0}\right\}(x-0{)}^k\dots 0\mathrm{点}\mathrm{ま}\mathrm{わ}\mathrm{り}\mathrm{の}\mathrm{テ}\mathrm{イ}\mathrm{ラ}\mathrm{ー}\mathrm{展}\mathrm{開}(Taylorseries)=\mathrm{マ}\mathrm{ク}\mathrm{ロ}\mathrm{ー}\mathrm{リ}\mathrm{ン}\mathrm{展}\mathrm{開}(Maclaurinexpansion)}\\ & =& {\displaystyle \frac{1}{0!}\left\{\left(\frac{{\mathrm{d}}^0}{\mathrm{d}{x}^0}{\mathrm{e}}^{Cx}\right){|}_{x=0}\right\}(x-0{)}^0{\displaystyle +\frac{1}{1!}\left\{\left(\frac{{\mathrm{d}}^1}{\mathrm{d}{x}^1}{\mathrm{e}}^{Cx}\right){|}_{x=0}\right\}(x-0{)}^1{\displaystyle +\frac{1}{2!}\left\{\left(\frac{{\mathrm{d}}^2}{\mathrm{d}{x}^2}{\mathrm{e}}^{Cx}\right){|}_{x=0}\right\}(x-0{)}^2{\displaystyle +\cdots }}}}\\ & =& {\displaystyle \frac{1}{0!}\left\{\left({\mathrm{e}}^{Cx}\right){|}_{x=0}\right\}(x-0{)}^0{\displaystyle +\frac{1}{1!}\left\{\left(C{\mathrm{e}}^{Cx}\right){|}_{x=0}\right\}(x-0{)}^1{\displaystyle +\frac{1}{2!}\left\{\left(C^2{\mathrm{e}}^{Cx}\right){|}_{x=0}\right\}(x-0{)}^2{\displaystyle +\cdots }}}}\\ & =& {\displaystyle \frac{1}{0!}1x^0{\displaystyle +\frac{1}{1!}C{x}^1{\displaystyle +\frac{1}{2!}{C}^2{x}^2{\displaystyle +\cdots \dots a^0=1}}}}\\ & =& {\displaystyle \frac{1}{0!}{C}^0{x}^0{\displaystyle +\frac{1}{1!}{C}^1{x}^1{\displaystyle +\frac{1}{2!}{C}^2{x}^2{\displaystyle +\cdots \dots a^0=1}}}}\\ & =& {\displaystyle \sum _{k=0}^{\mathrm{\infty }}\frac{(Cx{)}^k}{k!}}\end{array}$$ $$\begin{array}{rcl}{\displaystyle \sum _{k=0}^{\mathrm{\infty }}\frac{(Cx{)}^k}{k!}}& =& {\mathrm{e}}^{Cx}\end{array}$$