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^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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