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:
(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,
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(n2).
It does hold for infinite sums too: 1 + 1/2 + 1/4 + ... + 1/2i + ... = ∑i=0..∞ 1/2i = Θ(1).