I believe the equation used on the page is wrong. It states projected winnings as 1/2 x $2 + 1/4 * $4 + ... But you only have a 1/4 chance of winning $2, since have the time you get a heads on first flip, you also get a heads on the second flip, meaning you'll win more than $2.

The equation should be: 1/4 x $2 + 1/8 x $4 ...

Still goes to infinity, but at half the pace.