When to stop rolling a die in a game where 6 loses everything You play a game using a standard six-sided die. You start with 0 points. Before every roll, you decide whether you want to continue the game or end it and keep your points. After each roll, if you rolled 6, then you lose everything and the game ends. Otherwise, add the score from the die to your total points and continue/stop the game.
When should one stop playing this game? Obviously, one wants to maximize total score.
As I was asked to show my preliminary results on this one, here they are:
If we simplify the game to getting 0 on 6 and 3 otherwise, we get the following: 
$$\begin{align}
EV &= \frac{5}{6}3+\frac{25}{36}6+\frac{125}{216}9+\ldots\\[5pt]
&= \sum_{n=1}^{\infty}\left(\frac{5}{6}\right)^n3n
\end{align}$$ 
which is divergent, so it would make sense to play forever, which makes this similar to the St. Petersburg paradox. Yet I can sense that I'm wrong somewhere!

Why is your summation divergent?   For n sufficiently large, each term is less than 123456 * (11/12)^n, which certainly converges to something finite.   Particularly, look up "gradient series" in a book on engineering economics (time value of money).   There you will find that x/(1-x)^2 = x+2x^+3x^3+4x^4+...  Or you can get it by differentiating both sides of 1/(1-x) = Sum(i=0 to infinity)(x^i).  This would seem to indicate that your summation adds up to 90.    Of course, selecting a strategy that maximizes your EXPECTED payoff is no guarantee of success on any single game, but it is the best you can do if you play thousands of games.    
