Taylor's Theorem/One Variable/Statement of Theorem
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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:
\(\ds \map f x\) | \(=\) | \(\ds \frac 1 {0!} \map f a\) | ||||||||||||
\(\ds \) | \(\) | \(\, \ds + \, \) | \(\ds \frac 1 {1!} \paren {x - a} \map {f'} a\) | |||||||||||
\(\ds \) | \(\) | \(\, \ds + \, \) | \(\ds \frac 1 {2!} \paren {x - a}^2 \map {f} a\) | |||||||||||
\(\ds \) | \(\) | \(\, \ds + \, \) | \(\ds \cdots\) | |||||||||||
\(\ds \) | \(\) | \(\, \ds + \, \) | \(\ds \frac 1 {n!} \paren {x - a}^n \map {f^{\paren n} } a\) | |||||||||||
\(\ds \) | \(\) | \(\, \ds + \, \) | \(\ds R_n\) |
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:
- $\ds \map f x = \sum_{n \mathop = 0}^\infty \frac {\paren {x - a}^n} {n!} \map {f^{\paren n} } a$
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 $x \in \openint a b$:
\(\ds \map f x\) | \(=\) | \(\ds \frac 1 {0!} \map f \xi\) | ||||||||||||
\(\ds \) | \(\) | \(\, \ds + \, \) | \(\ds \frac 1 {1!} \paren {x - \xi} \map {f'} \xi\) | |||||||||||
\(\ds \) | \(\) | \(\, \ds + \, \) | \(\ds \frac 1 {2!} \paren {x - \xi}^2 \map {f} \xi\) | |||||||||||
\(\ds \) | \(\) | \(\, \ds + \, \) | \(\ds \cdots\) | |||||||||||
\(\ds \) | \(\) | \(\, \ds + \, \) | \(\ds \frac 1 {n!} \paren {x - \xi}^n \map {f^{\paren n} } \xi\) | |||||||||||
\(\ds \) | \(\) | \(\, \ds + \, \) | \(\ds R_n\) |
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 $\xi$.
Source of Name
This entry was named for Brook Taylor.
Sources
- 1972: Frank Ayres, Jr. and J.C. Ault: Theory and Problems of Differential and Integral Calculus (SI ed.): Chapter $53$: Maclaurin's and Taylor's Formulas with Remainders
- 2005: Roland E. Larson, Robert P. Hostetler and Bruce H. Edwards: Calculus (8th ed.): $\S 9.7$