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Appendix C

Closed Forms for Sums of Geometric Series

Summations of geometric series appear often in problems involving various transforms and in the analysis of linear systems. In this appendix we give derivations of closed-form formulas for infinite- and finite-length geometric series.

C.1 Infinite-Length Geometric Series

Consider the sum of infinite-length geometric series in the form

\begin{matrix} P = \sum_{n = 0}^{\infty} a^{n} & (C .1) \end{matrix}

$\begin{matrix} P = \sum_{n = 0}^{\infty} a^{n} & (C .1) \end{matrix}$

For the summation in Eqn. (C.1) to converge we must have |a| < 1. Assuming that is the case, let us write Eqn. (C.1) in open form:

\begin{array}{l} p = 1 + a + a^{2} + a^{3} + ... & (C .2) \end{array}

$\begin{array}{l} p = 1 + a + a^{2} + a^{3} + ... & (C .2) \end{array}$

Subtracting unity from both sides of Eqn. (C.2) we obtain

\begin{array}{l} p = - 1 + a + a^{2} + a^{3} + ... & (C .3) \end{array}

$\begin{array}{l} p = - 1 + a + a^{2} + a^{3} + ... & (C .3) \end{array}$

in which the terms on the right side of the equal sign have a common factor a .Factoring it out leads to

\begin{array}{l} P - 1 & = a (1 + a + a^{2} + a^{3} + ...) \\ = a p & (C .4) \end{array}

$\begin{array}{l} P - 1 & = a (1 + a + a^{2} + a^{3} + ...) \\ = a p & (C .4) \end{array}$

which can be solved for P to yield

\begin{array}{l} P = \frac{1}{1 - a}, & | a | < 1 & (C .5) \end{array}

$\begin{array}{l} P = \frac{1}{1 - a}, & | a | < 1 & (C .5) \end{array}$

C.2 Finite-Length Geometric Series

Consider the finite-length sum of a geometric series in the form

\begin{array}{l} Q = \sum_{n = 0}^{L} a^{n} & (C .6) \end{array}

$\begin{array}{l} Q = \sum_{n = 0}^{L} a^{n} & (C .6) \end{array}$

Unlike P of the previous section, the convergence of Q does not require |a| < 1. The only requirement is that |a| < ∞. Let us write Q in open form:

\begin{array}{l} Q = 1 + a + ... + a^{L} & (C .7) \end{array}

$\begin{array}{l} Q = 1 + a + ... + a^{L} & (C .7) \end{array}$

Subtracting unity from both sides of Eqn. (C.7) leads to

\begin{array}{l} Q - 1 = a + ... + a^{L} & (C .8) \end{array}

$\begin{array}{l} Q - 1 = a + ... + a^{L} & (C .8) \end{array}$

As before, the terms on the right side of the equal sign have a common factor a which can be factored out to yield

\begin{array}{l} Q - 1 & = a (1 + ... + a^{L - 1}) \\ = a (Q - a^{L}) & (C .9) \end{array}

$\begin{array}{l} Q - 1 & = a (1 + ... + a^{L - 1}) \\ = a (Q - a^{L}) & (C .9) \end{array}$

which can be solved for Q:

\begin{array}{l} Q = \frac{1 - a^{L + 1}}{1 - a} & (C .10) \end{array}

$\begin{array}{l} Q = \frac{1 - a^{L + 1}}{1 - a} & (C .10) \end{array}$

Consistency check: If L is increased, in the limit Q approaches P provided that |a| < 1, that is

\begin{array}{l} lim_{L \to \infty} [Q] = P & if & | a | < 1 & (C .11) \end{array}

$\begin{array}{l} lim_{L \to \infty} [Q] = P & if & | a | < 1 & (C .11) \end{array}$

C.3 Finite-Length Geometric Series (Alternative Form)

Consider an alternative form of the finite-length geometric series sum in which the lower limit is not equal to zero.

\begin{matrix} Q = \sum_{n = L_{1}}^{L_{2}} a^{n} & (C .12) \end{matrix}

$\begin{matrix} Q = \sum_{n = L_{1}}^{L_{2}} a^{n} & (C .12) \end{matrix}$

In order to obtain a closed form formula we will apply the variable change m = n − L1 with which Eqn. (C.12) becomes

\begin{array}{l} Q = \sum_{m = 0}^{L_{2} - L_{1}} a^{m +} L_{1} & (C .13) \end{array}

$\begin{array}{l} Q = \sum_{m = 0}^{L_{2} - L_{1}} a^{m +} L_{1} & (C .13) \end{array}$

which can be simplified as

\begin{array}{l} Q = a^{L_{1}} \sum_{m = 0}^{L_{2} - L_{1}} a^{m} & (C .14) \end{array}

$\begin{array}{l} Q = a^{L_{1}} \sum_{m = 0}^{L_{2} - L_{1}} a^{m} & (C .14) \end{array}$

The summation on the right side of Eqn. (C.14) is in the standard form of Eqn. (C.6), therefore

\begin{array}{l} Q = a^{L 1} (\frac{1 - a^{L 2 - L_{1} + 1}}{1 - a}) = \frac{a^{L_{1}} - a^{L_{2} + 1}}{1 - a} & (C .15) \end{array}

$\begin{array}{l} Q = a^{L 1} (\frac{1 - a^{L 2 - L_{1} + 1}}{1 - a}) = \frac{a^{L_{1}} - a^{L_{2} + 1}}{1 - a} & (C .15) \end{array}$

Consistency check: For L1 = 0 and L2 = L, Eqn. (C.15) reduces to Eqn. (C.10).

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Table of Contents for Appendix C Closed Forms for Sums of Geometric Series

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Closed Forms for Sums of Geometric Series

C.1 Infinite-Length Geometric Series

C.2 Finite-Length Geometric Series

C.3 Finite-Length Geometric Series (Alternative Form)

Table of Contents for
Appendix C Closed Forms for Sums of Geometric Series