I have the following problem:

Prove that iterative method:$$x\_{n+1} = \frac{x\_n(x\_n^2 + 3R)}{3x\_n^2 + R}$$ converges cubically to $\sqrt{R}$.

So, as I understand, I have to show two things:

* the method converges to $\sqrt{R}$
* $\lim\_{n\rightarrow\infty} |\frac{x\_{n+1} - \sqrt{R}}{(x\_n - \sqrt{R})^3}| > 0$

but I have no idea how to do it.