Calculus Cameron Geisler

\begin{align*} \boxed{ \begin{aligned} \frac{1}{1-x} \amp= \sum_{n=0}^{\infty} x^{n} \amp= 1 + x + x^{2} + x^{3} + \cdots \amp\quad -1 < x < 1 \\[0.6ex] e^{x} \amp= \sum_{n=0}^{\infty} \frac{x^{n}}{n!} \amp= 1 + x + \frac{x^{2}}{2!} + \frac{x^{3}}{3!} + \cdots \amp\quad -\infty < x < \infty \\[0.6ex] \sin x \amp= \sum_{n=0}^{\infty} (-1)^{n}\frac{x^{2n+1}}{(2n+1)!} \amp= x - \frac{x^{3}}{3!} + \frac{x^{5}}{5!} - \frac{x^{7}}{7!} + \cdots \amp\quad -\infty < x < \infty \\[0.6ex] \cos x \amp= \sum_{n=0}^{\infty} (-1)^{n}\frac{x^{2n}}{(2n)!} \amp= 1 - \frac{x^{2}}{2!} + \frac{x^{4}}{4!} - \frac{x^{6}}{6!} + \cdots \amp\quad -\infty < x < \infty \\[0.6ex] \ln(1+x) \amp= \sum_{n=0}^{\infty} (-1)^n \frac{x^{n+1}}{n+1} \amp= x - \frac{x^{2}}{2} + \frac{x^{3}}{3} - \frac{x^{4}}{4} + \cdots \amp\quad -1 < x \le 1 \\[0.6ex] \arctan x \amp= \sum_{n=0}^{\infty} (-1)^{n}\frac{x^{2n+1}}{2n+1} \amp= x - \frac{x^{3}}{3} + \frac{x^{5}}{5} - \frac{x^{7}}{7} + \cdots \amp\quad -1 \le x \le 1 \end{aligned}} \end{align*}

\begin{align*} \tan x \amp = x + \frac{x^3}{3} + \frac{2x^5}{15} + \frac{17x^7}{315} + \cdots, \quad \lvert x\rvert < \frac{\pi}{2} \\\\ (1+x)^n \amp = 1 + nx + \frac{n(n-1)}{2!}x^2 + \frac{n(n-1)(n-2)}{3!}x^3 + \cdots = \sum_{k=0}^{\infty} \binom{n}{k} x^k, \quad \lvert x\rvert < 1 \\\\ \sinh x \amp = x + \frac{x^3}{3!} + \frac{x^5}{5!} + \cdots = \sum_{n=0}^{\infty} \frac{x^{2n+1}}{(2n+1)!}, \quad -\infty < x < \infty \\\\ \cosh x \amp = 1 + \frac{x^2}{2!} + \frac{x^4}{4!} + \cdots = \sum_{n=0}^{\infty} \frac{x^{2n}}{(2n)!}, \quad -\infty < x < \infty \end{align*}