## Apr 27 Taylor Series to Memorize

There are four taylor series that you MUST memorize

MathJax TeX Test Page $$e^x (c = 0)$$ $$P_n(x) = 1 + x + \frac{x^2}{2} + \frac{x^3}{6} + ... + \frac{x^n}{n!}$$ $$=\sum_{n = 0}^{\infty}\frac{x^n}{n!}$$
$$\cos{(x)}$$ $$P_n(x) = 1 - \frac{x^2}{2} + \frac{x^4}{4!} - \frac{x^6}{6!} + \frac{x^8}{8!}...$$ $$=\sum_{n=0}^{\infty}(-1)^n\frac{x^{2n}}{(2n)!}$$
MathJax TeX Test Page $$\sin(x)$$ This is similar to cos $$\sum_{n = 0}^{\infty}(-1)^n \frac{x^{2n +1}}{(2n+1)!}$$
MathJax TeX Test Page $$\frac{1}{1-x}$$ Note that this is identical to $\frac{a}{1-r}$, the geometric series formula, with a = 1. For this reason, the sum is $$\sum_{n = 0}^{\infty}x^n$$
David Witten