There is a game of betting on the flip of a coin: you keep flipping the coin until you get a tail. If you get a tail on the first throw you get $2, if the tail is on the second throw you get $4, then $8 if on the third, and so on.

The question is, what amount would be fair payment to play this game? Your expected winnings are the sum of the gain at each toss, divided by its probability. This is: $2/2 + $4/4 + $8/8 + $16/16 + ... = $1 + $1 + $1 + $1 + ... The trouble is that this sum adds up to infinity. No-one would pay an infinite amount in order to play this game, as sooner or later a tail will appear and you win only a finite amount.