Then, if the payoff were constant, the total win would be because eventually the player wins times before the game ends.
Originally Posted by salohcin
Taking the geometric decrease of the payoff into account, the total win is where are the winning times. Note that is a geometric random variable of parameter (after k wins, the next win has probability to happen), and these random variables are independent.
If is a geometric r.v. with parameter , then one computes . Using this, we get (up to possible mistakes)
This is not very nice, but I'm not sure this can be turned to something nicer...