Taylor's Theorem/One Variable/Proof by Rolle's Theorem
Contents
Theorem
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 $\xi$.
Proof
Let the function $g$ be defined as:
- $g \left({t}\right) = R_n \left({t}\right) - \dfrac {\left({t - a}\right)^{n + 1} } {\left({x - a}\right)^{n + 1} } R_n \left({x}\right)$
Then:
- $g^{\left({k}\right)} \left({a}\right) = 0$
for $k = 0, \dotsc, n$, and $g \left({x}\right) = 0$.
Apply Rolle's Theorem successively to $g, g', \dotsc, g^{\left({n}\right)}$.
Then there exist:
- $\xi_1, \ldots, \xi_{n + 1}$
between $a$ and $x$ such that:
- $g' \left({\xi_1}\right) = 0, g'' \left({\xi_2}\right) = 0, \ldots, g^{\left({n + 1}\right)} \left({\xi_{n + 1} }\right) = 0$
Let $\xi = \xi_{n + 1}$.
Then:
- $0 = g^{\left({n + 1}\right)} \left({\xi}\right) = f^{\left({n + 1}\right)} \left({\xi}\right) - \dfrac {\left({n + 1}\right)!} {\left({x - a}\right)^{n + 1} } R_n \left({x}\right)$
and the formula for $R_n \left({x}\right)$ follows.
$\blacksquare$
Source of Name
This entry was named for Brook Taylor.
Sources
- 1977: K.G. Binmore: Mathematical Analysis: A Straightforward Approach ... (previous) ... (next): $\S 11.10$