Definition:Taylor Series/Remainder/Lagrange 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 Lagrange form of the remainder $R_n$ is given by:
 * $R_n = \dfrac {f^{\left({n + 1}\right)} \left({x^*}\right)} {\left({n + 1}\right)!} \left({x - \xi}\right)^{n + 1}$

where $x^* \in \left({\xi \,.\,.\, x}\right)$.

Also see

 * Definition:Cauchy Form of Remainder of Taylor Series