As part of the programming I do, I recently stumbled
into the sequence $\: \left\langle x_0,x_1,x_2,x_3,... \right\rangle \:$ defined as follows:
$x_{\hspace{0.01 in}0} = 1 \qquad$ and $\qquad$ for all non-negative integers $n$, $\;\;\; x_{\hspace{0.01 in}n+1}\:=\:\frac12 \cdot x_n \cdot (2-x_n)$
I am aware that $\: \left\langle x_0,x_1,x_2,x_3,... \right\rangle \:$ approaches $0$ from the right.
Is there a closed-form expression for $x_{\hspace{0.01 in}n}$?
If no, is there a good closed-form estimate of $x_{\hspace{0.01 in}n}$ for large $n\hspace{0.01 in}$?
(In particular, a better estimate than $0$.)