Section3.2Taylor's Theorem
Let's start by recalling the definition of a Taylor Series. Loosely speaking, if \(f\) is infinitely smooth (has infinitely many derivatives) then \begin{equation*} f(x) = \sum_{k=0}^\infty \frac{f^{(k)}(a)}{k!}(x-a)^k \end{equation*} in some neighborhood of \(x=a\). The power here is that Taylor Series allow for the approximation of smooth functions as polynomials.
Theorem3.2.2Taylor's Theorem
Let \(f\), \(f'\), \(f''\), …, \(f^{(n)}\) be continuous on near \(a\) and let \(f^{(n+1)}(x)\) exist for all \(x\) near \(a\). Then there is a number \(\xi\) between \(x\) and \(a\) such that \begin{gather*} f(x) = f(a) + \frac{f'(a)}{1} (x-a) + \frac{f''(a)}{2}(x-a)^2 + \cdots + \frac{f^{(n)}}{n!}(x-a)^n + R_n(x) \end{gather*} where the remainder function \(R_n(x)\) is given as \begin{gather*} R_n(x) = \frac{f^{(n+1)}(\xi)}{(n+1)!} (x-a)^{n+1} \end{gather*}