Formal Statement of Taylor's Theorem

GUIDE: Mathematics of the Discrete Fourier Transform (DFT) - Julius O. Smith III. Formal Statement of Taylor's Theorem

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NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW VERSION: "Mathematics of the Discrete Fourier Transform (DFT), with Audio Applications --- Second Edition", by Julius O. Smith III, W3K Publishing, 2007, ISBN 978-0-9745607-4-8. - Copyright © 2017-09-28 by Julius O. Smith III - Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

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Formal Statement of Taylor’s Theorem

Let $f(x)$ be continuous on a real interval $I$ containing $x_0$ (and $x$), and let $f^{(n)}(x)$ exist at $x$ and $f^{(n+1)}(\ be continuous for all $\. Then we have the following Taylor series expansion:


where $R_{n+1}(x)$ is called the remainder term. There exists $\ between $x$ and $x_0$ such that

In particular, if $\ in $I$, then

which is normally small when $x$ is close to $x_0$.

When $x_0=0$, the Taylor series reduces to what is called a Maclaurin series.

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