Definition:Taylor Series/Remainder

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

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

for some $n \in \N$.

The difference:
 * $\displaystyle \map {R_n} x = \map f x - \map {T_n} {\map f \xi} = \int_\xi^x \map {f^{\paren {n + 1} } } t \dfrac {\paren {x - t}^n} {n!} \rd t$

is known as the remainder of $\map T {\map f \xi}$ at $x$.