**NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW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 , the signal , , defines a

geometric sequence, i.e., each term is obtained by multiplying the previous term by a (complex) constant. Ageometric seriesis defined as thesumof a geometric sequence:

If , the sum can be expressed inclosed formas

Proof:We have