# 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 $\map T {\map f \xi}$ of $f$ about the point $\xi$:

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

Let $\map {T_n} {\map f \xi}$ be the Taylor polynomial:

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

for some $n \in \N$.

The difference:

$\displaystyle \map {R_n} x = \map f x - \map {T_n} {\map f \xi} = \int_\xi^x \map {f^{\paren {n + 1} } } t \dfrac {\paren {x - t}^n} {n!} \rd t$

is known as the remainder of $\map T {\map f \xi}$ 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.