Taylor's Theorem/One Variable/Statement of Theorem

Theorem
Let $f$ be a real function which is continuous on the closed interval $\closedint a b$ and $n + 1$ times differentiable on the open interval $\openint a b$.

Let $\xi \in \openint a b$.

Then, given any $x \in \openint a b$, 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:
 * $R_n = \dfrac 1 {\paren {n + 1}!} \paren {x - \xi}^{n + 1} \map {f^{\paren {n + 1} } } \eta$

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

The expression:
 * $\displaystyle \map f x = \sum_{n \mathop = 0}^\infty \frac {\paren {x - \xi}^n} {n!} \map {f^{\paren n} } \xi$

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