Taylor's Theorem

Theorem

Taylor's Theorem states that any infinitely differentiable function (including one where the derivative is $0$) 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 $n$ times differentiable on the open interval $\left({a..b}\right)$.

Let $\xi \in \left({a..b}\right)$.

Then, given any $x \in \left({a..b}\right)$:

	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle f \left({x}\right)\)	\(=\)	\(\displaystyle \frac 1 {0!} f \left({\xi}\right)\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(+\)	\(\displaystyle \frac 1 {1!} \left({x - \xi}\right) f^{\prime} \left({\xi}\right)\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(+\)	\(\displaystyle \frac 1 {2!} \left({x - \xi}\right)^2 f^{\prime \prime} \left({\xi}\right)\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(+\)	\(\displaystyle \ldots\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(+\)	\(\displaystyle \frac 1 {n!} \left({x - \xi}\right)^n f^{\left({n}\right)} \left({\xi}\right)\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(+\)	\(\displaystyle R_n\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)

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

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

where $\eta \in \R$ is some (at this point unspecified) real number such that $x \le \eta \le \xi$.

Note that when $n = 1$ Taylor's Theorem reduces to the Mean Value Theorem.

The expression:

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

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.