Definition:Taylor Series/Remainder

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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 $x$.


Lagrange Form

The Lagrange form of the remainder $R_n$ is given by:

$R_n = \dfrac {\map {f^{\paren {n + 1} } } {x^*} } {\paren {n + 1}!} \paren {x - \xi}^{n + 1}$

where $x^* \in \openint \xi x$.


Cauchy Form

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


Sources