二項級数
\begin{eqnarray*}
(a+x)^n&=&a^n+na^{n-1}x+\frac{n(n-1)}{2!}a^{n-2}x^2+\frac{n(n-1)(n-2)}{3!}a^{n-3}x^3+ \cdots\\
\frac{1}{1+x}&=&1-x+x^2-x^3+x^4- \cdots \left(-1 < x < 1\right)\\
\frac{1}{(1+x)^2}&=&1-2x+3x^2-4x^3+5x^4- \cdots \left(-1 < x < 1\right)\\
\frac{1}{(1+x)^3}&=&1-3x+6x^2-10x^3+15x^4- \cdots \left(-1 < x < 1\right)\\
\sqrt{1+x}&=&1+\frac{1}{2}x-\frac{1}{2 \cdot 4}x^2+\frac{1 \cdot 3}{2 \cdot 4 \cdot 6}x^3- \cdots \left(-1 < x \le 1\right)\\
\frac{1}{\sqrt{1+x}}&=&1-\frac{1}{2}x+\frac{1 \cdot 3}{2 \cdot 4}x^2-\frac{1 \cdot 3 \cdot 5}{2 \cdot 4 \cdot 6}x^3+ \cdots \left(-1 < x \le 1\right)\\
\sqrt[3]{1+x}&=&1+\frac{1}{3}x-\frac{2}{3 \cdot 6}x^2+\frac{2 \cdot 5}{3 \cdot 6 \cdot 9}x^3- \cdots \left(-1 < x \le 1\right)\\
\frac{1}{\sqrt[3]{1+x}}&=&1-\frac{1}{3}x+\frac{1 \cdot 4}{3 \cdot 6}x^2-\frac{1 \cdot 4 \cdot 7}{3 \cdot 6 \cdot 9}x^3+ \cdots \left(-1 < x \le 1\right)
\end{eqnarray*}
指数および対数関数
\begin{eqnarray*}
e^x&=&1+x+\frac{x^2}{2!}+\frac{x^3}{3!}+ \cdots \left(-\infty < x < \infty\right)\\
a^x&=&e^{x\ln a}=1+x\ln a+\frac{(x\ln a)^2}{2!}+\frac{(x\ln a)^3}{3!}+ \cdots \left(-\infty < x < \infty\right)\\
\ln (1+x)&=&x-\frac{x^2}{2}+\frac{x^3}{3}-\frac{x^4}{4}+ \cdots \left(-1 < x \le 1\right)\\
\ln x&=&\left(\frac{x-1}{x}\right)+\frac{1}{2}\left(\frac{x-1}{x}\right)^2+\frac{1}{3}\left(\frac{x-1}{x}\right)^3+ \cdots \left(\frac{1}{2} \le x\right)
\end{eqnarray*}
三角関数
\begin{eqnarray*}
\sin x&=&x-\frac{x^3}{3!}+\frac{x^5}{5!}-\frac{x^7}{7!}+ \cdots \left(-\infty < x < \infty\right)\\
\cos x&=&1-\frac{x^2}{2!}+\frac{x^4}{4!}-\frac{x^6}{6!}+ \cdots \left(-\infty < x < \infty\right)\\
\tan x&=&x+\frac{x^3}{3}+\frac{2x^5}{15}+\frac{17x^7}{315}+ \cdots \left(|x|<\frac{\pi}{2}\right)
\end{eqnarray*}
逆三角関数
\begin{eqnarray*}
\sin^{-1} x&=&x+\frac{1}{2} \cdot \frac{x^3}{3}+\frac{1\cdot 3}{2 \cdot 4} \cdot \frac{x^5}{5}+\frac{1\cdot 3 \cdot 5}{2 \cdot 4 \cdot 6} \cdot \frac{x^7}{7}+ \cdots \left(|x|<1\right)\\
\cos^{-1} x&=&\frac{\pi}{2}-\sin^{-1} x=\frac{\pi}{2}-\left( x+\frac{1}{2} \cdot \frac{x^3}{3}+\frac{1\cdot 3}{2 \cdot 4} \cdot \frac{x^5}{5}+\frac{1\cdot 3 \cdot 5}{2 \cdot 4 \cdot 6} \cdot \frac{x^7}{7}+ \cdots\right) \left(|x|<1\right)\\
\tan^{-1} x&=&\left\{\begin{array}{l}
\displaystyle \frac{\pi}{2}-\frac{1}{x}+\frac{1}{3x^3}-\frac{1}{5x^5}+ \cdots \left(1 \le x\right)\\
\displaystyle x-\frac{x^3}{3}+\frac{x^5}{5}-\frac{x^7}{7}+ \cdots \left(-1 < x < 1\right)\\
\displaystyle -\frac{\pi}{2}-\frac{1}{x}+\frac{1}{3x^3}-\frac{1}{5x^5}+ \cdots \left(x \le -1\right)\\
\end{array}\right .
\end{eqnarray*}
双曲線関数
\begin{eqnarray*}
\sinh x&=&x+\frac{x^3}{3!}+\frac{x^5}{5!}+\frac{x^7}{7!}+ \cdots \left(-\infty < x < \infty\right)\\
\cosh x&=&1+\frac{x^2}{2!}+\frac{x^4}{4!}+\frac{x^6}{6!}+ \cdots \left(-\infty < x < \infty\right)\\
\tanh x&=&x-\frac{x^3}{3}+\frac{2x^5}{15}-\frac{17x^7}{315}+ \cdots \left(|x|<\frac{\pi}{2}\right)
\end{eqnarray*}
逆双曲線関数
\begin{eqnarray*}
\sinh^{-1} x&=&\left\{\begin{array}{l}
\displaystyle \ln 2x+\left(\frac{1}{2} \cdot \frac{1}{2x^2}-\frac{1 \cdot 3}{2 \cdot 4} \cdot \frac{1}{4x^4}+\frac{1 \cdot 3 \cdot 5}{2 \cdot 4 \cdot 6} \cdot \frac{1}{6x^6}- \cdots \right) \left(1 \le x\right)\\
\displaystyle x-\frac{1}{2} \cdot \frac{x^3}{3}+\frac{1 \cdot 3}{2 \cdot 4} \cdot \frac{x^5}{5}-\frac{1 \cdot 3 \cdot 5}{2 \cdot 4 \cdot 6} \cdot \frac{x^7}{7}+ \cdots \left(-1 < x < 1\right)\\
-\ln |2x|-\left(\displaystyle \frac{1}{2} \cdot \frac{1}{2x^2}-\frac{1 \cdot 3}{2 \cdot 4} \cdot \frac{1}{4x^4}+\frac{1 \cdot 3 \cdot 5}{2 \cdot 4 \cdot 6} \cdot \frac{1}{6x^6}- \cdots \right) \left(x \le -1\right)\\
\end{array}\right .\\
\cosh^{-1} x&=&\left\{\begin{array}{l}
\ln 2x-\displaystyle \left(\frac{1}{2} \cdot \frac{1}{2x^2}-\frac{1 \cdot 3}{2 \cdot 4} \cdot \frac{1}{4x^4}+\frac{1 \cdot 3 \cdot 5}{2 \cdot 4 \cdot 6} \cdot \frac{1}{6x^6}- \cdots \right) \left(0 < \cosh^{-1}x, \hspace{3mm} 1 \le x\right)\\
-\ln 2x+\displaystyle \left(\frac{1}{2} \cdot \frac{1}{2x^2}-\frac{1 \cdot 3}{2 \cdot 4} \cdot \frac{1}{4x^4}+\frac{1 \cdot 3 \cdot 5}{2 \cdot 4 \cdot 6} \cdot \frac{1}{6x^6}- \cdots \right) \left(\cosh^{-1}x <0, \hspace{3mm} 1 \le x\right)\\
\end{array}\right .\\
\tanh^{-1} x&=&x+\frac{x^3}{3}+\frac{x^5}{5}+\frac{x^7}{7}+ \cdots \left(|x| \le 1\right)\\
\end{eqnarray*}