Then, if the payoff were constant, the total win would be

because eventually the player wins

times before the game ends.

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...