Maybe the clearest way to see it is that the expected number of boys is one (obviously--there's always exactly 1), and the expected number of girls is also one

Maybe the clearest way to see it is that the expected number of boys is one (obviously--there's always exactly 1), and the expected number of girls is also one (1/2 + 1/4 + ...) since the prob. of (at least) one girl is 1/2, and the prob. of a second girl is 1/4, and so on.