Similar to Gambler's Ruin, I am interested in calculating the number of trials accumulated until a continuously-reinvested, finite bankroll has reached exhaustion. I have come up with the formula below but am unsure of its soundness.
$$\displaystyle{p}={\sum_{{{i}={0}}}^{\infty}}{{b}{v}^{i}}$$
where ${p}$ is the sum of all expenditures, ${b}$ is the bankroll, and $v$ is the expected value of the game.
An example:
Player A has 5,000 units and decides to play a game with expected value of 0.65. What is the total number of units he will play before his bankroll is exhausted?
Plugging in the numbers, I get approximately 14,285. If Player A starts out with 5,000 units then he can expect to accumulate 14,285 trials before exhaustion if he reinvests his winnings each time.
My logic is that every trial has an expected value of 0.65 so Player A could expect, on average, to receive 0.65 units in exchange for his 1 unit each round (i.e. he has 5000 round 0, 3250 round 1, 2112.5 round 2, etc). Am I correct? I feel like my logic must be flawed because he is only betting 1 unit per round, not all units.