Question 18
Set $u_0 = \dfrac{1}{4},$ 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 $$
|u_k - L| \le \dfrac{1}{2^{1000}}?
$$
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 \dfrac{1}{2^{1000}}?
$$