Proof that Various Infinities Have a Finite Value

In the late 19th century, German mathematician Georg Cantor demonstrated that there are a variety of infinities of various sizes. Equations 1 and 2 below compare two different arbitrary infinities:

The infinity at equation 2 is n times bigger than the infinity at equation 1.

Srinivasa Ramanujan, an Indian mathematician (1887–1920), was cleverly able to extract finite values from various divergent infinite series. If having various infinities isn't weird enough, imagine various infinities having a finite value! Normally, the process of extracting a finite value from an infinity requires tackling that infinity with a unique set of steps. Thus it is not clear if all infinities have a finite value. Perhaps there are some that do and some that don't. Can mathematics provide any clues? Let's begin with an arbitrary function f that diverges to infinity:

What makes this particular infinity unique is not that variable N tends to infinity, but that N is multiplied by a coefficient alpha. This infinity is alpha times the size of a benchmark infinity that is simply N with an infinite limit. Below is another infinity (function s) with a coefficient of beta, where beta may not equal alpha:

The relationship between f and s is as follows:

Using some Ramanujanian algebra, we can solve s:

We can now solve f by making a substitution:

If coefficient k is finite and x doesn't equal 1, the solution to any arbitrary infinite function is finite. Otherwise it is infinite.

References:

1. Matson, John. 07/19/2007. Strange but True: Infinity Comes in Different Sizes. Scientific American

2. Dodds, Mark. 09/02/2018. The Ramanujan Summation: 1 + 2 + 3 + ⋯ + ∞ = -1/12? Cantor's Paradise