Number bigger than infinity

Obviously some infinities are "bigger" than others and you clearly know that. But your arguments hinge upon undefined concepts, which make them bad argumemts.
This is sort of like saying infinity + 1 or 2 times infinity are obviously bigger than infinity.

It doesn't work like that.
How can 2+2+2+... be a bigger infinity than 1+1+1+... where 2=1+1 so 2+2+2+...=(1+1)+(1+1)+(1+1)+...=1+1+1+...?
>we know that this infinite series >= to infinity but the wrong assumption here is that there isn't any number bigger than infinity.

There isn't any number equal to infinity either in the reals. When we say that a series "sums to infinity" what we mean is that the sum doesn't exist, but it's a short hand for the sum failing to exist in a particular way.

The rigorous definition of a sum -> infinity is that if you pick any large positive number, eventually the sequence of partial sums gets bigger than that and stays bigger than that. The partial sums are finite numbers, and so is the large positive number you're comparing them to.

Your proofs are doing multiple operations that are not defined in the reals. They are not valid. And we already know that Taylor series don't converge for all x.
A lot of what you did is wrong and/or ill defined.
A divergent series does not converge to infinity. It just doesn't converge. Infinity is not a number. You cannot reorder the terms of a divergent sequence as you wish. You cannot reorder them at all. There's no such thing as division by zero.

If you are interested in infinities however there is much to look at. These are called ordinals and they are related to power sets. The first infinity is the countable-listable-numerable. The racionals and the integers are countable and they have the same size. The intuition is that with you could write them out, one by one, if you had infinite time. The integers is easy. The racionals is more ingeneous ( Cantor's diagonalization ). Then you have the uncountable infinity ( 1st ordinal ). The reals and irrationals are uncountable. The idea of ordinals is that the power set ( set of subsets) of an infinite set is strictly bigger, even if infinite, that the set.
Infinity is not a number, it is meaningless to act like it is. It is a concept
this is all completely meaningless because everything that you are doing is based upon vague ideas that do not have real definitions.
1/0 is just not possible. If you say it has multiple values it leads to instant contradictoon..