Taylor's Theorem/One Variable/Statement of Theorem/Also presented as

Taylor's Theorem in One Variable 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 $x \in \openint a b$:

$\ds \map f x$

$=$

$\ds \frac 1 {0!} \map f \xi$

$\ds $

$\, \ds + \, $

$\ds \frac 1 {1!} \paren {x - \xi} \map {f'} \xi$

$\ds $

$\, \ds + \, $

$\ds \frac 1 {2!} \paren {x - \xi}^2 \map {f} \xi$

$\ds $

$\, \ds + \, $

$\ds \cdots$

$\ds $

$\, \ds + \, $

$\ds \frac 1 {n!} \paren {x - \xi}^n \map {f^{\paren n} } \xi$

$\ds $

$\, \ds + \, $

$\ds R_n$

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 $\xi$.

This entry was named for Brook Taylor.