Definition:Taylor Series/Remainder

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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 $x$.


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:

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

where $\eta \in \closedint \xi x$.


Sources