### Geometric Series

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

when

In other words, the geometric series is guaranteed to be summable when , and in that case, the sum is given by . On the other hand, if , we can rewrite as to obtain

which is summable when . Thus, is a valid closed-form sum whether or not is less than or greater than 1. When , it is the sum of the causal geometric series in powers of . When , it is the sum of the causal geometric series in powers of , or, an anticausal geometric series in (negative) powers of .

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