What I don't understand is how you get conditional convergence from ... ... if there was a 2^-n chance of getting 2^n utility and a 2^-n/n chance of getting -2^n/n utility (before normalizing)

I just noticed, that should have been 2^-n chance of getting 2^n/n utility and a 2^-n chance of getting -2^n/n

Anyway, 2^-n*2^n/n = 1/n, so the expected utility from that possibility is 1/n, so you get an unbounded expected utility. Do it with negative too, and you get conditionally converging expected utility.