Disclaimer, I'm a computer person, not a math person, so surely I'm not going to use the proper math lingo. I apologize in advance. Feel free to edit to make this question more readable to others. This is a 2 part question:
What is the probability of achieving a goal that has a probability of $1/n$, with $n$ tries, as $n$ goes to infinity?
The actual question I'm trying to solve is related to a hypothetical poker game, and I believe its answer hinges on the answer to question 1 above. The game is as follows: Player 1 $(P_1)$ begins with a score of $x$, Player 2 $(P_2)$ begins with a score of $N*x$. Both players "go all in" on the flip of a coin comparison, and wager the amount of the lower player's score. If $P_1$ gets down to 0, he starts with $x$ again. We continue until $P_2$ has 0. What is the expected number of times this game can be played until $P_2$ hits 0, in terms of $x$ and $N$?
I wrote a simple computer program to illustrate the game, and played it with x = 1 and N = 100. After 20 or so tries, here was one result that was short enough to paste here:
Starting Position - $P_1:1, P_2:100$
Hand Count: 1 $P_1:2, P_2:99$
Hand Count: 2 $P_1:4, P_2:97$
Hand Count: 3 $P_1:0, P_2:101$
Hand Count: 4 $P_1:0, P_2:102$
Hand Count: 5 $P_1:2, P_2:101$
Hand Count: 6 $P_1:4, P_2:99$
Hand Count: 7 $P_1:0, P_2:103$
Hand Count: 8 $P_1:2, P_2:102$
Hand Count: 9 $P_1:4, P_2:100$
Hand Count: 10 $P_1:0, P_2:104$
Hand Count: 11 $P_1:2, P_2:103$
Hand Count: 12 $P_1:4, P_2:101$
Hand Count: 13 $P_1:8, P_2:97$
Hand Count: 14 $P_1:16, P_2:89$
Hand Count: 15 $P_1:32, P_2:73$
Hand Count: 16 $P_1:64, P_2:41$
Hand Count: 17 $P_1:105, P_2:0$
$P_1$ stated with 1. $P_1$ Ended with 105.
$P_2$ started with 100. $P_2$ Ended with 0.
Total Hands played until $P_1$ busts $P_2$: 17
Total amount $P_1$ invested: 5
Notes: Hand #4 shows the result after Player 1 performed a "rebuy" of 1 and immediately lost. Hand #5 show the result after $P_1$ performed a rebuy of 1 and won, etc. In order for Player 1 to ever win this game, he must win 7 in a row, and I'm thinking he has about 128 tries to do it before he then has to hit 8 in a row, then 256 tries to achive the $1/256$ before getting 512 tries at $1/512$, etc.