part of page 762

$$\frac{1}{1-x}=\sum_{n=0}^\infty x^n$$
$$e^x=\sum_{n=0}^\infty \frac{x^n}{n!}$$
$$\sin x=\sum_{n=0}^\infty (-1)^n\frac{x^{2n+1}}{(2n+1)!}$$
$$\cos x=\sum_{n=0}^\infty (-1)^n\frac{x^{2n}}{(2n)!}$$
Note: $\sin 0=0$ and $\cos 0=1$
$$\tan^{-1} x=\sum_{n=0}^\infty (-1)^n\frac{x^{2n+1}}{2n+1}$$
$$\ln(1+x)=\sum_{n=1}^\infty (-1)^{n-1}\frac{x^n}{n}$$
$$(1+x)^{k}=\sum_{n=0}^{\infty}\left(\begin{array}{c}
k\\
n\end{array}\right)x^{n}$$
$$\left(\begin{array}{c}
k\\
n\end{array}\right)=\frac{k(k-1)(k-2)\cdots(k-n+1)}{n!}$$