Geometric Series

GUIDE: Mathematics of the Discrete Fourier Transform (DFT) - Julius O. Smith III. Geometric Series

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NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW VERSION: "Mathematics of the Discrete Fourier Transform (DFT), with Audio Applications --- Second Edition", by Julius O. Smith III, W3K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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Geometric Series

Recall that for any compex number $z_1\, the signal $x(n)\, $n=0,1,2,\, defines a geometric sequence, i.e., each term is obtained by multiplying the previous term by a (complex) constant. A geometric series is defined as the sum of a geometric sequence:

\

If $z_1\, the sum can be expressed in closed form as
\

Proof: We have
\


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