# 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:

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

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

### Remainder

Consider the Taylor series expansion $T \left({f \left({\xi}\right)}\right)$ of $f$ about the point $\xi$:

$\displaystyle \sum_{n \mathop = 0}^\infty \frac {\left({x - \xi}\right)^n} {n!} f^{\left({n}\right)} \left({\xi}\right)$

Let $T_n \left({f \left({\xi}\right)}\right)$ be the Taylor polynomial:

$\displaystyle \sum_{n \mathop = 0}^n \frac {\left({x - \xi}\right)^n} {n!} f^{\left({n}\right)} \left({\xi}\right)$

for some $n \in \N$.

The difference:

$\displaystyle R_n \left({x}\right) = f \left({x}\right) - T_n \left({f \left({\xi}\right)}\right) = \int_\xi^x f^{\left({n + 1}\right)} \left({t}\right) \dfrac {\left({x - t}\right)^n} {n!} \, \mathrm d t$

is known as the remainder of $T \left({f \left({\xi}\right)}\right)$ at $x$.

## Source of Name

This entry was named for Brook Taylor.

## Historical Note

The first proof for the convergence of a Taylor series was provided by Augustin Louis Cauchy.

He used the Cauchy Form of the remainder, showing that it converges to zero.