Taylor's Theorem/One Variable/Statement of Theorem

From ProofWiki
Jump to navigation Jump to search

Theorem

Let $f$ be a real function which is:

of differentiability class $C^n$ on the closed interval $\closedint a x$

and:

at least $n + 1$ times differentiable on the open interval $\openint a x$.

Then:

\(\displaystyle \map f x\) \(=\) \(\displaystyle \frac 1 {0!} \map f a\)
\(\displaystyle \) \(\) \(\, \displaystyle + \, \) \(\displaystyle \frac 1 {1!} \paren {x - a} \map {f'} a\)
\(\displaystyle \) \(\) \(\, \displaystyle + \, \) \(\displaystyle \frac 1 {2!} \paren {x - a}^2 \map {f''} a\)
\(\displaystyle \) \(\) \(\, \displaystyle + \, \) \(\displaystyle \cdots\)
\(\displaystyle \) \(\) \(\, \displaystyle + \, \) \(\displaystyle \frac 1 {n!} \paren {x - a}^n \map {f^{\paren n} } a\)
\(\displaystyle \) \(\) \(\, \displaystyle + \, \) \(\displaystyle R_n\)

where $R_n$ (sometimes denoted $E_n$) is known as the error term or remainder, and can be presented in one of $2$ forms:

Lagrange Form
$R_n = \dfrac {\map {f^{\paren {n + 1} } } \xi} {\paren {n + 1}!} \paren {x - a}^{n + 1}$

for some $\xi \in \openint a x$.

Cauchy Form
$R_n = \dfrac {\map {f^{\paren {n + 1} } } \xi \paren {x - \xi}^n} {n!} \paren {x - a}$

for some $\xi \in \openint a x$.


Taylor Series Expansion

The expression:

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

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


Also presented as

Taylor's Theorem in One Variable can also be presented in a form like this or similar:

Let $f$ be a real function which is at least $n + 1$ times differentiable on the open interval $\openint a b$.

Let $\xi$ be a real number in $\openint a b$.

Then for a given $x \in \openint a b$:

\(\displaystyle \map f x\) \(=\) \(\displaystyle \frac 1 {0!} \map f \xi\)
\(\displaystyle \) \(\) \(\, \displaystyle + \, \) \(\displaystyle \frac 1 {1!} \paren {x - \xi} \map {f'} \xi\)
\(\displaystyle \) \(\) \(\, \displaystyle + \, \) \(\displaystyle \frac 1 {2!} \paren {x - \xi}^2 \map {f''} \xi\)
\(\displaystyle \) \(\) \(\, \displaystyle + \, \) \(\displaystyle \cdots\)
\(\displaystyle \) \(\) \(\, \displaystyle + \, \) \(\displaystyle \frac 1 {n!} \paren {x - \xi}^n \map {f^{\paren n} } \xi\)
\(\displaystyle \) \(\) \(\, \displaystyle + \, \) \(\displaystyle R_n\)

where $E_n$ satisfies:

$E_n = \dfrac 1 {\paren {n + 1}!} \paren {x - \xi}^{n + 1} \map {f^{\paren {n + 1} } } \eta$

for some $\eta$ between $x$ and $\xi$.


Source of Name

This entry was named for Brook Taylor.


Sources