# 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:

\(\displaystyle \map f x\) | \(=\) | \(\displaystyle \frac 1 {0!} \map f a\) | |||||||||||

\(\displaystyle \) | \(\) | \(\, \displaystyle + \, \) | \(\displaystyle \frac 1 {1!} \paren {x - a} \map {f'} a\) | ||||||||||

\(\displaystyle \) | \(\) | \(\, \displaystyle + \, \) | \(\displaystyle \frac 1 {2!} \paren {x - a}^2 \map {f''} a\) | ||||||||||

\(\displaystyle \) | \(\) | \(\, \displaystyle + \, \) | \(\displaystyle \cdots\) | ||||||||||

\(\displaystyle \) | \(\) | \(\, \displaystyle + \, \) | \(\displaystyle \frac 1 {n!} \paren {x - a}^n \map {f^{\paren n} } a\) | ||||||||||

\(\displaystyle \) | \(\) | \(\, \displaystyle + \, \) | \(\displaystyle R_n\) |

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

- 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.): $\S 9.7$