##### Geometric Given $x \in(-1,1)$ $ \begin{aligned} \frac{1}{1-x} &=1+x+x^{2}+x^{3}+x^{4}+\ldots \\ &=\sum_{n=0}^{\infty} x^{n} \end{aligned} $ #### Exponential and Logs ##### Exponential Given $x \in\mathbb{R}$ $ \begin{aligned} e^{x} &=1+x+\frac{x^{2}}{2 !}+\frac{x^{3}}{3 !}+\frac{x^{4}}{4 !}+\ldots \\ &=\sum_{n=0}^{\infty} \frac{x^{n}}{n !} \end{aligned} $ ##### Log Given $x \in(-1,1]$ $ \begin{aligned} \ln (1+x) &=x-\frac{x^{2}}{2}+\frac{x^{3}}{3}-\frac{x^{4}}{4}+\frac{x^{5}}{5}-\ldots \\ &=\quad \sum_{n=1}^{\infty}(-1)^{(n-1)} \frac{x^{n}}{n} \stackrel{\text { or }}{=} \sum_{n=1}^{\infty}(-1)^{n+1} \frac{x^{n}}{n} \end{aligned} $ #### Trig ##### Sine Given $x \in\mathbb{R}$ $ \begin{aligned} \cos x &=1-\frac{x^{2}}{2 !}+\frac{x^{4}}{4 !}-\frac{x^{6}}{6 !}+\frac{x^{8}}{8 !}-\cdots \\ &=\sum_{n=0}^{\infty}(-1)^{n} \frac{x^{2 n}}{(2 n) !} \end{aligned} $ ##### Cosine Given $x \in\mathbb{R}$ $ \begin{aligned} \sin x &=x-\frac{x^{3}}{3 !}+\frac{x^{5}}{5 !}-\frac{x^{7}}{7 !}+\frac{x^{9}}{9 !}-\ldots \\ &=\quad \sum_{n=1}^{\infty}(-1)^{(n-1)} \frac{x^{2 n-1}}{(2 n-1) !} \stackrel{\text { or }}{=} \sum_{n=0}^{\infty}(-1)^{n} \frac{x^{2 n+1}}{(2 n+1) !} \end{aligned} $ ##### Inverse Tangent Given $x \in[-1,1]$ $\begin{aligned} \tan ^{-1} x &=x-\frac{x^{3}}{3}+\frac{x^{5}}{5}-\frac{x^{7}}{7}+\frac{x^{9}}{9}-\ldots \\ &=\sum_{n=1}^{\infty}(-1)^{(n-1)} \frac{x^{2 n-1}}{2 n-1} \stackrel{\text { or }}{=} \sum_{n=0}^{\infty}(-1)^{n} \frac{x^{2 n+1}}{2 n+1} \end{aligned}$