Two Nonstandard Infinite Geometric Series (visual proof)

My previous wordless Wednesday video showing the sum of the powers of 1/7 generated some interesting questions about using the same technique for nonstandard geometric series. Here we include a short, animated visual proof demonstrating the infinite sum of the powers of 2/7 and 3/11; from these dissections, you should be able to generate a similar circular diagram for any rational number ratio r=a/b where the infinite sum will be between 0 and 1. These two series were explicitly mentioned by James Tanton and Dr. Barker.
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[DrBarker]

Wow, these are very cool! I can see quite clearly how these would generalise to any rational common ratio r ≤ 1/2, which I wouldn't have thought was possible. But I think for e.g. r = 2/3, you may need to tweak this approach slightly, as you can't have 2/3 in the centre and also 2/3 on the outside. I guess there should still be a nice way of doing it though. So perhaps there's even more to think about for this kind of dissection proof!

[mike_the_tutor]

This construction is possible whenever 0 <= r < 1, a = (x/y)(1 - r), y is a positive natural number, and x is a nonnegative natural number, such that x <= y. The outermost circle has area 1. The first inner circle has area r. x is the number of shaded wedges. y is the total number of wedges, both shaded and unshaded. The outermost shaded wedges, together, have an area of a. In every case, the sum will be x/y.

EDIT: Should be 0 <= r < 1 instead of |r| < 1, since r is an area. Also, x <= y.

[ccbgaming]

Wow it’s beautiful like Denis Cyplenkov

[scmtuk]

Did you know that the sum of the powers of 69/420 = 69/351?

This is because, obviously the sum of the powers of a/b = a/(b-a)

Sum of the powers of 3/11 = 3/(11-3) = 3/8
Sum of the powers of 2/7 = 2/(7-2) = 2/5

And therefore, the sum of the powers of x/(x+1) = x
This is because if the sum of the powers of a/b = a/(b-a) then the sum of the powesr of x/(x+1) = x/(x-x+1) = x/1 = x

For example the sum of the powers of 3/4 = 3.

If a/b are equal, then the sum of the powers of a/b = a/(b-a) = a/(a-a) = a/0 = undefined.

This is because you're literally asking for the sum of the powers of 1. I.e., the value of 1 + 1 + 1 + 1 + 1 .... which is .... well, basically ∞

But what if the fraction is improper?
The sum of the powers of 3/2 (i.e. 1.5) would therefore be 3/(2-3) = 3/-1 = -3, right? Well, obviously it's infinity, but .... what if it isn't?

And what about negative fractions?
The sum of the powers of -2/3 = -2/(3--2) = -2/5, right? Well, in this case, yes.

[zapazap]

"Nonstandard" has a technical meaning in math that does not apply here.