Appendix B. Closed Form of a Geometric
Series
In the literature of digital signal processing,
we often encounter geometric series expressions like
or
Equation B2
Unfortunately, many authors make a statement
like "and we know that," and drop Eqs. (B1) or (B2) on the unsuspecting reader who's
expected to accept these expressions on faith. Assuming that you
don't have a Ph.D. in mathematics, you may wonder exactly what
arithmetic sleight of hand allows us to arrive at Eqs. (B1) or (B2). To answer this
question, let's consider a general expression for a geometric
series such as
Equation B3
where n, N, and
p are integers and a and r
are any constants. Multiplying Eq. (B3) by r, gives us
Equation B4
Subtracting Eq. (B4) from Eq. (B3) gives the expression
or
Equation B5
So here's what we're after. The closed form of the series is
Equation B6
(By closed form, we mean taking an infinite
series and converting it to a simpler mathematical form without the
summation.) When a = 1, Eq. (B6) validates Eq. (B1). We can quickly
verify Eq. (B6) with an
example. Letting N = 5, p = 0, a = 2, and r = 3, for example, we can create the
following list:
n

ar^{n} = 2 · 3^{n}

0

2 · 3^{0} = 2

1

2 · 3^{1} = 6

2

2 · 3^{2} = 18

3

2 · 3^{3} = 54

4

2 · 3^{4} = 162


The sum of this column is

Plugging our example N, p, a, and r values into Eq. (B6),
Equation B7
which equals the sum of the rightmost column in
the list above.
As a final step, the terms of our earlier Eq. (B2) are in the form of
Eq. (B6) as p = 0, a = 1, and r = e^{–}^{j}^{2p}^{m}^{/}^{N}^{.}^{[]} So plugging
those terms from Eq.
(B2) into Eq. (B6)
gives us
^{[]} From the math identity a^{xy} = (a^{x})^{y}, we can say e^{–j}^{2p}^{nm}^{/}^{N} = (e^{–j}^{2p}^{m}^{/}^{N})^{n}, so r = e^{–}^{j}^{2p}^{m}^{/}^{N}.
Equation B8
confirming Eq. (B2).
