# Taylor's Theorem

## Theorem

Every infinitely differentiable function can be approximated by a series of polynomials.

### One Variable

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, given any $x \in \left({a \,.\,.\, b}\right)$, there exists some $\eta \in \R: x \le \eta \le \xi$ or $\xi \le \eta \le x$ such that:

\(\displaystyle f \left({x}\right)\) | \(=\) | \(\displaystyle \frac 1 {0!} f \left({\xi}\right)\) | |||||||||||

\(\displaystyle \) | \(\) | \(\, \displaystyle + \, \) | \(\displaystyle \frac 1 {1!} \left({x - \xi}\right) f^{\prime} \left({\xi}\right)\) | ||||||||||

\(\displaystyle \) | \(\) | \(\, \displaystyle + \, \) | \(\displaystyle \frac 1 {2!} \left({x - \xi}\right)^2 f^{\prime \prime} \left({\xi}\right)\) | ||||||||||

\(\displaystyle \) | \(\) | \(\, \displaystyle + \, \) | \(\displaystyle \cdots\) | ||||||||||

\(\displaystyle \) | \(\) | \(\, \displaystyle + \, \) | \(\displaystyle \frac 1 {n!} \left({x - \xi}\right)^n f^{\left({n}\right)} \left({\xi}\right)\) | ||||||||||

\(\displaystyle \) | \(\) | \(\, \displaystyle + \, \) | \(\displaystyle R_n\) |

where $R_n$ (sometimes denoted $E_n$) is known as the **error term**, and satisfies:

- $R_n = \dfrac 1 {\left({n + 1}\right)!} \left({x - \xi}\right)^{n + 1} f^{\left({n + 1}\right)} \left({\eta}\right)$

Note that when $n = 0$ Taylor's Theorem reduces to the Mean Value Theorem.

The expression:

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

where $n$ is taken to the limit, is known as the Taylor series expansion of $f$ about $\xi$.

## Source of Name

This entry was named for Brook Taylor.

## Historical Note

Taylor's Theorem, as applied to an analytic function, was estabished by Carl Friedrich Gauss in $1831$, but he never got round to publishing this work.