Geometric Series

Recall that for any complex number $z_1\in{\bf C}$, the signal $x(n)\isdef z_1^n$, $n=0,1,2,\ldots\,$, defines a geometric sequence, i.e., each term is obtained by multiplying the previous term by a (complex) constant. A geometric series is the sum of a geometric sequence:

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

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

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

Proof: We have

$$\begin{eqnarray*} S_N(z_1) &\isdef & 1 + z_1 + z_1^2 + z_1^3 + \cdots + z_1^{N-1... ...) &=& 1-z_1^N \\ \;\implies\;S_N(z_1) &=& \frac{1-z_1^N}{1-z_1} \end{eqnarray*}$$