e^{x} = \sum^{\infin}_{n=0} \frac{x^n}{n!} = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \cdots\quad\text{ for all } x\! \log(1-x) = - \sum^{\infin}_{n=1} \frac{x^n}n\quad\text{ for } |x| < 1 \log(1+x) = \sum^\infin_{n=1} (-1)^{n+1}\frac{x^n}n\quad\text{ for } |x| < 1 \frac{1}{1-x} = \sum^\infin_{n=0} x^n\quad\text{ for }|x| < 1\! (1+x)^\alpha = \sum_{n=0}^\infty {\alpha \choose n} x^n\quad\text{ for all }|x| < 1 \text{ and all complex } \alpha\! {\alpha\choose n} = \prod_{k=1}^n \frac{\alpha-k+1}k = \frac{\alpha(\alpha-1)\cdots(\alpha-n+1)}{n!} \sin x = \sum^{\infty}_{n=0} \frac{(-1)^n}{(2n+1)!} x^{2n+1} = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \cdots\quad\text{ for all } x\! \cos x = \sum^{\infin}_{n=0} \frac{(-1)^n}{(2n)!} x^{2n} = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \cdots\quad\text{ for all } x\! \tan x = \sum^{\infin}_{n=1} \frac{B_{2n} (-4)^n (1-4^n)}{(2n)!} x^{2n-1} = x + \frac{x^3}{3} + \frac{2 x^5}{15} + \cdots\quad\text{ for }|x| < \frac{\pi}{2}\! \sec x = \sum^{\infin}_{n=0} \frac{(-1)^n E_{2n}}{(2n)!} x^{2n}\quad\text{ for }|x| < \frac{\pi}{2}\! \arcsin x = \sum^{\infin}_{n=0} \frac{(2n)!}{4^n (n!)^2 (2n+1)} x^{2n+1}\quad\text{ for }|x| \le 1\! \arccos x ={\pi\over 2}-\arcsin x={\pi\over 2}- \sum^{\infin}_{n=0} \frac{(2n)!}{4^n (n!)^2 (2n+1)} x^{2n+1}\quad\text{ for }|x| \le 1\! \arctan x = \sum^{\infin}_{n=0} \frac{(-1)^n}{2n+1} x^{2n+1}\quad\text{ for }|x| \le 1, x\not=\pm i\! \sinh x = \sum^{\infin}_{n=0} \frac{x^{2n+1}}{(2n+1)!} = x + \frac{x^3}{3!} + \frac{x^5}{5!} + \cdots\quad\text{ for all } x\! \cosh x = \sum^{\infin}_{n=0} \frac{x^{2n}}{(2n)!} = 1 + \frac{x^2}{2!} + \frac{x^4}{4!} + \cdots\quad\text{ for all } x\! \tanh x = \sum^{\infin}_{n=1} \frac{B_{2n} 4^n (4^n-1)}{(2n)!} x^{2n-1} = x-\frac{1}{3}x^3+\frac{2}{15}x^5-\frac{17}{315}x^7+\cdots \quad\text{ for }|x| < \frac{\pi}{2}\! \mathrm{arcsinh} (x) = \sum^{\infin}_{n=0} \frac{(-1)^n (2n)!}{4^n (n!)^2 (2n+1)} x^{2n+1}\quad\text{ for }|x| \le 1\! \mathrm{arctanh} (x) = \sum^{\infin}_{n=0} \frac{x^{2n+1}}{2n+1} \quad\text{ for }|x| \le 1, x\not=\pm 1\!