I have been explaining the Riemann Rearrangement Theorem to a friend of mine, and they feel as though the definition of series using partial sums "doesn't work" for conditionally convergent sequences. I understand how they feel: the RRT feels counter-intuitive, and so the result should be denied as a contradiction and the partial-sum definition rejected.
However, the definition of a series as
$$\sum_{k=1}^{\infty} a_k = \lim_{n\to\infty} \left(\sum_{k=1}^n a_k\right)$$
feels like the most natural approach to take. If I wanted to add infinitely many numbers together by hand, this is how I would have to do it.
Is there an alternative (but not equivalent) definition of an infinite sum that agrees with the partial-sum definition on absolutely convergent sequences, but where the RRT doesn't hold?
My guess is that if you define some value for an infinite series agreeing on absolutely convergent sequences then this method of definition must imply the RRT, but I don't see how that proof could go.