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.