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Geometric Series

Recall that for any complex number $ z_1\in{\bf C}$ , the signal

$\displaystyle x(n)\isdef z_1^n,\quad n=0,1,2,\ldots,
$

defines a geometric sequence, i.e., each term is obtained by multiplying the previous term by the (complex) constant $ z_1$ . A geometric series is the sum of a geometric sequence:

$\displaystyle S_N(z_1) \isdef \sum_{n=0}^{N-1}z_1^n = 1 + z_1 + z_1^2 + z_1^3 + \cdots + z_1^{N-1}
$

If $ z_1\neq 1$ , the sum can be expressed in closed form:

$\displaystyle \zbox {S_N(z_1) = \frac{1-z_1^N}{1-z_1}}$   $\displaystyle \mbox{($z_1\neq 1$)}$



Proof: We have

\begin{eqnarray*}
S_N(z_1) &\isdef & 1 + z_1 + z_1^2 + z_1^3 + \cdots + z_1^{N-1} \\
z_1 S_N(z_1) &=& \qquad\!\! z_1 + z_1^2 + z_1^3 + \cdots + z_1^{N-1} + z_1^N \\
\,\,\Rightarrow\,\,
S_N(z_1) - z_1 S_N(z_1) &=& 1-z_1^N \\
\,\,\Rightarrow\,\,S_N(z_1) &=& \frac{1-z_1^N}{1-z_1}.
\end{eqnarray*}

When $ z_1=1$ , $ S_N(1)=N$ , by inspection of the definition of $ S_N(z_1)$ .


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``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 © 2014-04-21 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
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