Definition:Taylor Series

Definition
Let $f$ be a real function which is smooth on the open interval $\openint a b$.

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

Then the Taylor series expansion of $f$ about the point $\xi$ is:
 * $\ds \sum_{n \mathop = 0}^\infty \frac {\paren {x - \xi}^n} {n!} \map {f^{\paren n} } \xi$

It is not necessarily the case that this power series is convergent with sum $\map f x$.

Also see

 * Taylor's Theorem