[gjulianm]

The fact that it's a shorthand means that it's not in the definition, just a convention. In fact, nowhere in my studies I saw anyone using 'division' when working explicitly with fields in algebra, it's always multiplicative inverse.

By the way, you cannot do the addition like normal on that series, you don't end up at anything. You can say it diverges and its limit is +infinity (not aleph0, aleph0 is used for cardinality of sets so I think no one would use it as the result of a divergent series). When I said it 'broke math' what I meant was that, in the same way that OP did, it is a way to assign values to things that usually don't have one. I know it does not actually break math.