sinh(x)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)

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

$f(x)=\sinh \left(x\right)$のマクローリン展開(0点まわりのテイラー展開)

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