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)^n

So...

ex=1+x+x22!+x33!+x44!+...cos⁡x=1−x22!+x44!−x66!+⋯sin⁡x=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*}