Geometric Series

The essence of the situation can be illustrated using a simple geometric series. Let $R$ be any real (or complex) number. Then we have

$$\frac{1}{1-R} \eqsp 1 + R + R^2 + R^3 + \cdots \quad < \infty$$   when$$\quad\vert R\vert<1.$$

In other words, the geometric series $1 + R + R^2 + R^3 + \cdots$ is guaranteed to be summable when $\vert R\vert<1$ , and in that case, the sum is given by $1/(1-R)$ . On the other hand, if $\vert R\vert>1$ , we can rewrite $1/(1-R)$ as $-R^{-1}/(1-R^{-1})$ to obtain

$$\frac{1}{1-R} \eqsp \frac{-R^{-1}}{1-R^{-1}} \eqsp -R^{-1}\left[1 + R^{-1} + R^{-2} + R^{-3} + \cdots \right]$$

which is summable when $\vert R\vert>1$ . Thus, $1/(1-R)$ is a valid closed-form sum whether or not $\vert R\vert$ is less than or greater than 1. When $\vert R\vert<1$ , it is the sum of the causal geometric series in powers of $R$ . When $\vert R\vert>1$ , it is the sum of the causal geometric series in powers of $R^{-1}$ , or, an anticausal geometric series in (negative) powers of $R$ .