eを底とした指数凾数のマクローリン展開

a点まわりのテイラー展開

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

マクローリン展開(0点まわりのテイラー展開)

$$\begin{array}{rcl}f(x)& =& {\sum _{k=0}^{\mathrm{\infty }}\frac{f^{(k)}(a)}{x!}(x-a{)}^k|}_{a=0}\\ & =& \sum _{k=0}^{\mathrm{\infty }}\frac{f^{(k)}(0)}{x!}(x-0{)}^k\\ & =& \sum _{k=0}^{\mathrm{\infty }}\frac{f^{(k)}(0)}{x!}(x{)}^k\\ & =& \frac{1}{0!}{f}^{(0)}(0)x^0+\frac{1}{1!}{f}^{(1)}(0)x+\frac{1}{2!}{f}^{(2)}(0)x^2+\cdots \end{array}$$

$f(x)=e^x$のマクローリン展開(0点まわりのテイラー展開)

$$\begin{array}{rclrc}{e}^x& =& \frac{1}{0!}{f}^{(0)}(0)x^0& +\frac{1}{1!}{f}^{(1)}(0)x& +\frac{1}{2!}{f}^{(2)}(0)x^2+\cdots \\ & =& \frac{1}{0!}\left(e^0\right)x^0& +\frac{1}{1!}\left(e^0\right)x& +\frac{1}{2!}\left(e^0\right)x^2\\ & & +\frac{1}{3!}\left(e^0\right)x^3& +\frac{1}{4!}\left(e^0\right)x^4& +\frac{1}{5!}\left(e^0\right)x^5\\ & & +\frac{1}{6!}\left(e^0\right)x^6& +\frac{1}{7!}\left(e^0\right)x^7& +\frac{1}{8!}\left(e^0\right)x^8+\cdots \\ & =& \frac{1}{0!}\cdot 1\cdot x^0& +\frac{1}{1!}\cdot 1\cdot x& +\frac{1}{2!}\cdot 1\cdot x^2\\ & & +\frac{1}{3!}\cdot 1\cdot x^3& +\frac{1}{4!}\cdot 1\cdot x^4& +\frac{1}{5!}\cdot 1\cdot x^5\\ & & +\frac{1}{6!}\cdot 1\cdot x^6& +\frac{1}{7!}\cdot 1\cdot x^7& +\frac{1}{8!}\cdot 1\cdot x^8+\cdots \\ & =& \frac{1}{0!}{x}^0+\frac{1}{1!}{x}^1& +\frac{1}{2!}{x}^2+\frac{1}{3!}{x}^3& +\frac{1}{4!}{x}^4+\frac{1}{5!}{x}^5+\frac{1}{6!}{x}^6+\frac{1}{7!}{x}^7+\frac{1}{8!}{x}^8+\cdots \\ & =& 1+x+\frac{1}{2!}{x}^2& +\frac{1}{3!}{x}^3+\frac{1}{4!}{x}^4& +\frac{1}{5!}{x}^5+\frac{1}{6!}{x}^6+\frac{1}{7!}{x}^7+\frac{1}{8!}{x}^8+\cdots \end{array}$$