Formal Statement of Taylor's Theorem

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Formal Statement of Taylor's Theorem

Let be continuous on a real interval containing (and ), and let $f^{(n)}(x)$ exist at and $f^{(n+1)}(\xi)$ be continuous for all $\xi\in I$ . Then we have the following Taylor series expansion:

$\begin{eqnarray*} f(x) = f(x_0) &+& \frac{1}{1}f^\prime(x_0)(x-x_0) \\ [10pt] &... ...&+& \frac{1}{n!}f^{(n+1)}(x_0)(x-x_0)^n\\ [10pt] &+& R_{n+1}(x) \end{eqnarray*}$

where $R_{n+1}(x)$ is called the remainder term. Then Taylor's theorem [61, pp. 95-96] provides that there exists some $\xi$ between and such that

$\displaystyle R_{n+1}(x) = \frac{f^{(n+1)}(\xi)}{(n+1)!}(x-x_0)^{n+1}.$

In particular, if $\vert f^{(n+1)}\vert\leq M$ in

, then

$\displaystyle R_{n+1}(x) \leq \frac{M \vert x-x_0\vert^{n+1}}{(n+1)!}$

which is normally small when

is close to

When , the Taylor series reduces to what is called a Maclaurin series [54, p. 96].

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Next: Weierstrass Approximation Theorem

written by Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.

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Mathematics of the DFT
Taylor Series Expansions
Formal Statement of Taylor's Theorem