Assume that Jack goes to a casino. He plays n games. Every time he wins a game he gets 100$ and every time he looses a game he looses 50$. The probability that he wins i-th game is 1/i. In other words, the probabilities of winning first five games are 1,1/2,1/3,1/4,1/5, respectively.
Can we compute the expected amount of money he has won or lost after he
has played n games? An easier version will be how can we compute the expected amount after he has played 5 games.
Thanks in advance