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The essence of the situation can be illustrated using a simple
geometric series. Let
be any real (or complex) number. Then we
have
![$\displaystyle \frac{1}{1-R} \eqsp 1 + R + R^2 + R^3 + \cdots \quad < \infty$](img1075.png)
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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