Definition:Taylor Series

Definition
Let $f$ be a real function which is smooth on the open interval $\left({a \,.\,.\, b}\right)$.

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

Then the Taylor series expansion of $f$ about the point $\xi$ is:
 * $\displaystyle \sum_{n \mathop = 0}^\infty \frac {\left({x - \xi}\right)^n} {n!} f^{\left({n}\right)} \left({\xi}\right)$

It is not necessarily the case that this power series is convergent with sum $f \left({x}\right)$.

Also see

 * Taylor's Theorem