Definition:Taylor Polynomial

Definition
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, the polynomial $T_n$ defined by:
 * $\displaystyle T_n\left({x}\right) = \sum_{i \mathop = 0}^i \frac {\left({x - \xi}\right)^i} {i!} f^{\left({i}\right)} \left({\xi}\right)$

is known as the Taylor polynomial of degree $n$ for $f$ about $\xi$.

Also see

 * Taylor's Theorem