A *geometric sum* is a sum in which each term is a constant factor
larger than the previous term. (Or each term can be a constant factor smaller
than the previous term.) For example, here is a common geometric sum:

- 1 + 2 + 4 + 8 + … + 2
^{n}= 2^{n+1}- 1.

- 1 + x + x
^{2}+ x^{3}+ … + x^{n}= (x^{n+1}- 1)/(x-1) unless x=1.

- ∑
_{i=0..n}x^{i}= (x^{n+1}-1)/(x-1).

(We can prove this by multiplying both sides by (x-1) and simplifying.)

For analysis of algorithms, the important thing is that, for any x≠ 1, any geometric sum is proportional to its largest term. For example, for x> 1,

- ∑
_{i=0..n}x^{i}= Θ(x^{n}).

In words, **a geometric sum is proportional to its largest term.**

This fact is useful in analyzing algorithms, especially divide-and-concur algorithms.

Note that this is not true for all kinds of sums. For example, 1+2+...+n = Θ(n^{2}).

It does hold for infinite sums too: 1 + 1/2 + 1/4 + ... + 1/2^{i} + ... = ∑_{i=0..∞} 1/2^{i} = Θ(1).

- Section 1.3.1 of GoodrichAndTomassia