Suppose x is a positive number less than 1. What is the sum of the positive powers of x?
For example, suppose x=12.
The case where x=12 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. 0.5=0.12, 4=1002, 16=100002 and 1017=17.
The above trick works for the inverse of any positive integer. Suppose x=117.
We can generalize to any denominator d.
The relationship holds even when d is not an integer. Let r=1d.
This is the equation for a geometric series.
Nit: strictly speaking this only applies when |r|<1
(Which nicely corresponds to any base |d|>1. 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.