I am looking at the probability of losing $x$ games in a row, in a game where the probability of winning is $1/x$. (For example, if this is a fair casino game, what is the probability of losing $x$ before winning it back)
I have calculated this to be $(\frac{x}{x+1})^x$, and I have seen in excel that as $x$ increases, this number approaches about 0.368.
I am interested in how to calculate theoretically this assymptote, particularly because I am interested in also working out the assymptote for the function $(\frac{x}{x+1})^{(x/z)}$ where $z$ is a number greater than 1. (This situation looks at unfair casino games)
Edit:
Having thought further, I have deduced my first question is $\frac{1}{e}$, however I'm not sure how to deduce my second question that involves $z$. Thanks