I learned about the Riemann rearrangement theorem recently and I'm trying to develop an intuition as to why commutativity breaks down for conditionally convergent series.
I understand the technique used in the theorem, but it just seems really odd that commutativity breaks down to me. It doesn't happen for other properties -- associativity holds for convergent series and commutativity holds for absolutely convergent series. What makes this particular property for this particular kind of convergent series "special" in this way?
I'm aware of this question: why does commutativity of addition fail for infinite sums? but the answers haven't been helpful for me. JiK's answer is "why would it?", and goes on to talk about why you can't apply rules to infinite series, but this seems erroneous because associativity holds for convergent series and commutativity holds for absolutely convergent series. Fly by Night just explains conditionally vs absolutely convergent, josh314 says it applies to finite sums only which isn't true, and Denis just explains the theorem again, and Barry Cipra seems to have a similar kind of argument as JiK's and problematic for similar reasons.
Is there a good way to understand why this is happening intuitively? or is this the wrong way to think about it and I should just accept that it's happening even though it's unintuitive? It's hard for me to just let it go without an intution because it seems like math starts to "break" here.. the theorem is sound but this property no longer holds in this case, which is really strange to me
Does anyone know of any resources that go into depth on this kind of question?