Question 8
Set $u_0 = \frac{1}{4}$, and for $k \ge 0$ let $u_{k+1}$ be determined by the recurrence $u_{k+1} = 2u_k - 2u_k^2$. This sequence tends to a limit; call it $L$. What is the least value of $k$ such that
$$
|u_k - L| \le \frac{1}{2^{1000}}?
$$
$$
|u_k - L| \le \frac{1}{2^{1000}}?
$$