# Definition:Taylor Polynomial

## Definition

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 $\xi \in \openint a b$.

The polynomial $T_n$ defined as:

$\ds \map {T_n} x = \sum_{i \mathop = 0}^i \frac {\paren {x - \xi}^i} {i!} \map {f^{\paren i} } \xi$

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

That is, a Taylor polynomial is a Taylor series taken for $n$ initial terms.

## Source of Name

This entry was named for Brook Taylor.