GeometricSums

Geometric sums

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 + … + 2ⁿ = 2ⁿ⁺¹ - 1.

And, more generally,

1 + x + x² + x³ + … + xⁿ = (xⁿ⁺¹ - 1)/(x-1) unless x=1.

Using sum notation:

∑_i=0..n xⁱ = (xⁿ⁺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 xⁱ = Θ(xⁿ).

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).

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references:

• Section 1.3.1 of GoodrichAndTomassia