Is an infinite sum of rationals isomorphic with a regularly converging sequence of rationals (something along the lines of 1/2, 5/8, 11/16, etc.), where each rational in the sequence is the sum of all the addends up until then? I agree it is probably worth putting up a different definition anyway. I'm not sure I'll be able to do that for a little bit, since I haven't studied real analysis yet, but if you want to do that sooner, go for it. This is a fun conversation!