Taylor's theorem TaylorsTheorem 2001-11-08 01:00:39 2006-06-23 14:02:33 Theorem \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} \usepackage{graphicx} \usepackage{xypic} \section{Taylor's Theorem} Let $f$ be a function which is defined on the interval $(a,b)$ and suppose the $n$th derivative $f^{(n)}$ exists on $(a,b)$. Then for all $x$ and $x_0$ in $(a,b)$, $$ R_n(x) = \frac{f^{(n)}(y)}{n!}(x-x_0)^n $$ with $y$ strictly between $x$ and $x_0$ ($y$ depends on the choice of $x$). $R_n(x)$ is the $n$th remainder of the Taylor series for $f(x)$.