# Taylor theorem

$LaTeX: \textstyle \mbox{Let } f:(a-h, a+h) \to R \mbox{ be in } C^{n+1}.$ Then, for $LaTeX: \textstyle x \in (a, a+h),$

$LaTeX: f(x) = f(a) + \frac{[f^1 (a)] [x-a] + \dots + [f^n (a)] [(x-a)^n]}{n!} + R_n(x, a),$

where $LaTeX: \textstyle R_n(x, a),$ called the remainder, is given by the integral:

$LaTeX: \int_{a}^{x} \frac{(x - t)^n}{n!} f^{(n+1)} (t)\, dt.$

This extends to multivariate functions and is a cornerstone theorem in nonlinear programming. Unfortunately, it is often misapplied as an approximation by dropping the remainder, assuming that it goes to zero as $LaTeX: \textstyle x \to a.$ (See Myths and Counter Examples in Mathematical Programming to avoid misconception.)