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.

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Nit: strictly speaking this only applies when 

(Which nicely corresponds to any base . Hm. I wonder if this works in e.g. negabinary?)

It does work for negative bases. Representation of a number in any base is in essence a sum of base powers multiplied by coefficients. The geometric series just has all coefficients equal to 1 after the radix point (and a 1 before it, if we start addition from the 0th power).

Are these slips or am I misunderstanding the notation?

They are mistakes. Fixed. Thanks.