Chinese Children

There are a billion Chinese couples. They won't stop having children until they have a boy. Then they always stop. So five hundred million couples have one child, 250 million have two children etc. If you can sum the appropriate series you can tell me how many children will be born. If you can't there is another way. A logical way. What I like about this problem is that it shows how you can use probability problems to sum infinite series rather than the other way around. I doubt I am the first person to think of this but definitely never saw the idea elswhere.


Any way in this example, what is the answer, what is the series, and how can you solve it without summing the series? (Note. To be rigorous I would have had to state the number couples as approaching infinity.)