Geometric Series

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

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

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:

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

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

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

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