Taylor Series Taylor, a person, once said that f(x)=∑n=0∞(1n!)(dnfdxn)∣a⋅(x−a)nf(x) = \sum_{n=0}^{\infty}{\left(\frac{1}{n!}\right)\left(\frac{d^n f}{dx^n}\right)}|_a · (x - a)^nf(x)=n=0∑∞(n!1)(dxndnf)∣a⋅(x−a)n So... ex=1+x+x22!+x33!+x44!+...cosx=1−x22!+x44!−x66!+⋯sinx=x−x33!+x55!−x77!+⋯\begin{gather*} e^x=1+x+\frac{x^2}{2!}+\frac{x^3}{3!}+\frac{x^4}{4!}+...\\ \cos x=1-\frac{x^2}{2!}+\frac{x^4}{4!}-\frac{x^6}{6!}+\cdots\\ \sin x=x-\frac{x^3}{3!}+\frac{x^5}{5!}-\frac{x^7}{7!}+\cdots \end{gather*}ex=1+x+2!x2+3!x3+4!x4+...cosx=1−2!x2+4!x4−6!x6+⋯sinx=x−3!x3+5!x5−7!x7+⋯