Definition:Taylor Series/Remainder

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 Taylor series expansion $T \left({f \left({\xi}\right)}\right)$ of $f$ about the point $\xi$:
 * $\displaystyle \sum_{n \mathop = 0}^\infty \frac {\left({x - \xi}\right)^n} {n!} f^{\left({n}\right)} \left({\xi}\right)$

Let $T_n \left({f \left({\xi}\right)}\right)$ be the Taylor polynomial:
 * $\displaystyle \sum_{n \mathop = 0}^n \frac {\left({x - \xi}\right)^n} {n!} f^{\left({n}\right)} \left({\xi}\right)$

for some $n \in \N$.

The difference:
 * $\displaystyle R_n \left({x}\right) = f \left({x}\right) - T_n \left({f \left({\xi}\right)}\right) = \int_\xi^x f^{\left({n + 1}\right)} \left({t}\right) \dfrac {\left({x - t}\right)^n} {n!} \, \mathrm d t$

is known as the remainder of $T \left({f \left({\xi}\right)}\right)$ at $x$.