Commonly Used Taylor Series

About e

e^x=1+x+\frac{x^2}{2!}+\frac{x^3}{3!}+\frac{x^4}{4!}+\dots, x\in \R

\frac{1}{1-x}=1+x+x^2+x^3+x^4+\dots, x\in(-1,1)

\ln(1+x)=x-\frac{x^2}{2}+\frac{x^3}{3}-\frac{x^4}{4}+\dots, x\in(-1,1]

\ln\frac{1+x}{1-x}=2(x+\frac{x^3}{3}+\frac{x^5}{5}+\frac{x^7}{7}+\dots), x\in(-1,1)

This formula has a faster convergence speed and a larger range than ln(1+x), thus more often used.

About trigonometric function

\sin x=x-\frac{x^3}{3!}+\frac{x^5}{5!}-\frac{x^7}{7!}+\dots,x\in \R

\cos x=1-\frac{x^2}{2!}+\frac{x^4}{4!}-\frac{x^6}{6!}+\dots,x\in \R

\arctan x=x-\frac{x^3}{3}+\frac{x^5}{5}-\frac{x^7}{7}+\dots, x\in[-1,1]