a点まわりのテイラー展開
\begin{eqnarray}
f(x) &=& \sum_{k=0}^{\infty}\frac{f^{(k)}(a)}{x!}(x-a)^k
\;\ldots\;a点まわりのテイラー展開
\\&=& \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+\dotsb
\\&&\;\ldots\;f^{(n)}(x): f(x)のn階微分
\end{eqnarray}
マクローリン展開(0点まわりのテイラー展開)
\begin{eqnarray}
f(x) &=& \left.\sum_{k=0}^{\infty}\frac{f^{(k)}(a)}{x!}(x-a)^k\right|_{a=0}
\\&=& \sum_{k=0}^{\infty}\frac{f^{(k)}(0)}{x!}(x-0)^k
\\&=& \sum_{k=0}^{\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{eqnarray}
\(f(x)=\sinh{\left(x\right)}\)のマクローリン展開(0点まわりのテイラー展開)
\begin{eqnarray}
\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!}\left\{\sinh{(0)}\right\}x^0&+\frac{1}{1!}\left\{\cosh{(0)}\right\}x &+\frac{1}{2!}\left\{\sinh{(0)}\right\}x^2
\\&&+\frac{1}{3!}\left\{\cosh{(0)}\right\}x^3&+\frac{1}{4!}\left\{\sinh{(0)}\right\}x^4&+\frac{1}{5!}\left\{\cosh{(0)}\right\}x^5
\\&&+\frac{1}{6!}\left\{\sinh{(0)}\right\}x^6&+\frac{1}{7!}\left\{\cosh{(0)}\right\}x^7&+\frac{1}{8!}\left\{\sinh{(0)}\right\}x^8
+\cdots
\;\ldots\;\href{https://shikitenkai.blogspot.com/2021/04/cosh-sinh.html}{\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
\;\ldots\;\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{eqnarray}
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