I have the recursive formula:
\begin{align}
x_1&=1-\frac{1}{e} \\
x_{n+1}&=1-(1-x_n)^{1/x_n}
\end{align}
Is there any way to write this as an explicit formula? I've tried writing the exact forms of the first few terms but they get complicated very quickly and I can't see a pattern. I'm pretty sure it approaches $1$. Anyone know how to convert it, if it's even possible?
Some people have pointed out that this may be an XY problem, and you may be right. I'm essentially trying to find out two things about this equation:
a) is this function increasing (and does it approach 1)?
b) If it is increasing, at what n does xn pass a certain value (e.g., 0.9)
I thought converting to a function f(x) would be the easiest way to do this as I could then just plug in values and use a limit to find the answers I want, but I think I'm wrong about that, so any other ideas would be great.
This is essentially a probability equation I've set up with probability of an event xn changing for each n, so if the function is increasing it should approach one, although proving this would be nice too.