Exchanging limits and expectations is not always valid (eg http://www.stanford.edu/class/msande322/docs/app_B.pdf‎ http://www.stanford.edu/class/msande322/docs/

Exchanging limits and expectations is not always valid (eg http://www.stanford.edu/class/msande322/docs/app_B.pdf‎ http://www.stanford.edu/class/msande322/docs/app_B.pdf‎). In this case the expected gain is not convergent in the limit. See https://news.ycombinator.com/item?id=6387344 https://news.ycombinator.com/item?id=6387344 This is one of my favourite problems too :)