Question

Salut, mă puteți ajuta La problema 27...?

Answer 1

$x_{n+1} = x_n\sqrt{1-x_n^2},\quad x_0 \in(0,1) \\ \\ L = \lim\limits_{n\to \infty}x_n = \lim\limits_{n\to \infty}x_{n+1} = \lim\limits_{n\to \infty} \Big[x_n\sqrt{1-x_n^2}\Big] = \\ \\ =\lim\limits_{n\to \infty} (x_n) \sqrt{1- (\lim\limits_{n\to \infty}x_n)^2} = \\ \\ = L\sqrt{1-L^2} \\ \\ L = L\sqrt{1-L^2} \Big|^2 \Rightarrow L^2 = L^2(1-L^2) \Rightarrow L^2 = L^2-L^4 \Rightarrow \\ \\ \Rightarrow L^4 = 0 \Rightarrow \boxed{L = 0}$