Taylor's Theorem/One Variable/Statement of Theorem

Theorem
Let $f$ be a real function which is continuous on the closed interval $\left[{a \,.\,.\, b}\right]$ and $n + 1$ times differentiable on the open interval $\left({a \,.\,.\, b}\right)$.

Let $\xi \in \left({a \,.\,.\, b}\right)$.

Then, given any $x \in \left({a \,.\,.\, b}\right)$, there exists some $\eta \in \R: x \le \eta \le \xi$ or $\xi \le \eta \le x$ such that:

where $R_n$ (sometimes denoted $E_n$) is known as the error term, and satisfies:
 * $R_n = \dfrac 1 {\left({n + 1}\right)!} \left({x - \xi}\right)^{n + 1} f^{\left({n + 1}\right)} \left({\eta}\right)$

Note that when $n = 0$ Taylor's Theorem reduces to the Mean Value Theorem.

The expression:
 * $\displaystyle f \left({x}\right) = \sum_{n \mathop = 0}^\infty \frac {\left({x - \xi}\right)^n} {n!} f^{\left({n}\right)} \left({\xi}\right)$

where $n$ is taken to the limit, is known as the Taylor series expansion of $f$ about $\xi$.