Proving that a recursively defined sequence converges, Newton's method

Question

A sequence is defined recursively in the following manner:

$$ x_0 = 2, x_{n+1} = x_n - \frac{x_n^2 -3}{2x_n}, n = 0, 1, 2, ...$$

Prove that the sequence converges. What does the sequence converge to?

Since the sequence is basically Newton's method for the function $f(x) = x^2 -3$, the sequence must converge to either $\sqrt3$ or -$\sqrt3$, but I'm not sure which one of them. I tried proving that the sequence converges using mathematical induction, but that didn't get me anywhere.

Answer 1

Can you prove that $x_n>0$ implies $x_{n+1}>0$, to rule out negative limits for $x_n$?

Answer 2

write your term in the form $$x_{n+1}=\frac{1}{2}\left(x_n+\frac{3}{x_n}\right)$$ and by $AM-GM$ we get $$\frac{1}{2}\left(x_n+\frac{3}{x_n}\right)\geq \sqrt{3}$$

Answer 3

Note that if $x_{n+1} =\frac{1}{2}\left(x_n+\frac{a}{x_n}\right) $ then $x_{n+1}-\sqrt{a} =\frac{1}{2}\left(x_n-2\sqrt{a}+\frac{a}{x_n}\right) =\frac1{2\sqrt{x_n}}(x_n-\sqrt a)^2 $ so that once $x_n$ is close enough to $\sqrt a$, the sequence will converge.