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}
\(f(x)=e^{x}\)のマクローリン展開(0点まわりのテイラー展開)
\begin{eqnarray}
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{eqnarray}
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