Definition:Taylor Series/Remainder/Cauchy Form

Definition
Let $f$ be a real function which is smooth on the open interval $\openint a b$.

Let $\xi \in \openint a b$.

Consider the remainder of the Taylor series at $x$:
 * $\ds \map {R_n} x = \int_\xi^x \map {f^{\paren {n + 1} } } t \dfrac {\paren {x - t}^n} {n!} \rd t$

The Cauchy form of the remainder $R_n$ is given by:
 * $R_n = \dfrac {\paren {x - \eta}^n} {n!} \paren {x - \xi} \map {f^{\paren {n + 1} } } \eta$

where $\eta \in \closedint \xi x$.

Also see

 * Definition:Lagrange Form of Remainder of Taylor Series