Taylor's Theorem/One Variable/Statement of Theorem

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:

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 $a$.