1 billion[(1/2)*1 + (1/4)*2 + (1/8)*3 + ...]


From calculus we can identify the series in brackets as one half times the first derivative of the geometric series evaluated at 1/2, so its value is

.5*(1-1/2)^-2 = 2. So the answer is 2 billion.


Suppose there was a benevolent casino with a fair coin flip game which costs nothing to play, and we get paid a dollar for every head, and when we get a tail we also get paid a dollar but then the game terminates. The above series would represent the expected value for this game if we play until we flip a tail since half the time we would win a dollar, 1/4 of the time we win 2 dollars, etc. Now if this game had cost us a dollar per flip and only paid off on a head, and we decided to play until we lost a flip, the game would have an expectation of 0 regardless of how long we play since the expectation on each flip is 0. We would always lose the last bet. But in our benevolent game, instead of losing a dollar on the last bet we win a dollar, so this game has an expectation 2 dollars higher than 0 or 2 dollars.