Taylor's Theorem

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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:

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

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

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


Note that when $n = 1$ 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.