**NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW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 TheoremLet be continuous on a real interval containing (and ), and let exist at and be continuous for all . Then we have the following Taylor series expansion:

where is called the remainder term. There exists between and such that

In particular, if in , then

which is normally small when is close to .When , the Taylor series reduces to what is called a Maclaurin series.