Definition:Taylor Series/Remainder/Cauchy Form

Definition
Let $f$ be a real function which is smooth on the open interval $\left({a \,.\,.\, b}\right)$.

Let $\xi \in \left({a \,.\,.\, b}\right)$.

Consider the remainder of the Taylor series at $x$:
 * $\displaystyle R_n \left({x}\right)= \int_\xi^x f^{\left({n + 1}\right)} \left({t}\right) \dfrac {\left({x - t}\right)^n} {n!} \, \mathrm d t$

The Cauchy form of the remainder $R_n$ is given by:
 * $R_n = \dfrac {\left({x - x^*}\right)^n} {n!} \left({x - \xi}\right) f^{\left({n + 1}\right)} \left({x^*}\right)$

where $x^* \in \left[{\xi \,.\,.\, x}\right]$.

Also see

 * Definition:Lagrange Form of Remainder of Taylor Series