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.