Taylor and Maclaurin Series
$ f(x) = \sum_{n=0}^\infty\cfrac{f^{(n)}(a)}{n!}(x-a)^n\\ \qquad\quad= f(a) + \cfrac{f'(a)}{1!}(x-a) + \cfrac{f''(a)}{2!}(x-a) + \cfrac{f'''(a)}{3!}(x-a)+\cdots $
$ f(x) = \sum_{n=0}^\infty\cfrac{f^{(n)}(0)}{n!}(x-0)^n \\\qquad\quad = f(a) + \cfrac{f^{\prime}(0)}{1!}(x-0) + \cfrac{f^{\prime\prime}(0)}{2!}(x-0) + \cfrac{f^{\prime\prime\prime}(0)}{3!}(x-0)+\cdots $
$ \cfrac{1}{1-x} = \sum_{n=0}^\infty{x^n} = 1+x+x^2+x^3+\cdots\qquad R = 1\\\,\\e^x = \sum_{n=0}^\infty\cfrac{x^n}{n!} = 1 + \cfrac{x}{1!} + \cfrac{x^2}{2!} + \cfrac{x^3}{3!}+\cdots\qquad R = \infty \\\,\\sin\,x = \sum_{n=0}^\infty(-1)^n\cfrac{x^{2n+1}}{(2n+1)!} = x - \cfrac{x^3}{3!} + \cfrac{x^5}{5!} - \cfrac{x^7}{7!}+\cdots\qquad R = \infty\\\,\\cos\,x = \sum_{n=0}^\infty(-1)^n\cfrac{x^{2n}}{(2n)!} = 1 - \cfrac{x^2}{2!} + \cfrac{x^4}{4!} - \cfrac{x^6}{6!}+\cdots\qquad R = \infty\\\,\\tan^{-1}\,x = \sum_{n=0}^\infty(-1)^n\cfrac{x^{2n+1}}{(2n+1)} = x - \cfrac{x^3}{3} + \cfrac{x^5}{5} - \cfrac{x^7}{7}+\cdots\qquad R = 1\\\,\\ln(1+x) = \sum_{n=1}^\infty(-1)^{n-1}\cfrac{x^{n}}{n} = x - \cfrac{x^2}{2} + \cfrac{x^3}{3} - \cfrac{x^4}{4}+\cdots\qquad R = 1 $