# 14

Suppose is a positive number less than 1. What is the sum of the positive powers of ?

For example, suppose .

The case where is intuitively obvious in base-10, but it's even more intuitively obvious in base-2. I will use a subscript to indicate base e.g. , , and .

The above trick works for the inverse of any positive integer. Suppose .

We can generalize to any denominator .

The relationship holds even when is not an integer. Let .

This is the equation for a geometric series.

New Comment