The question: Consider a function such that $f(x)$, $f'(x)$, and $f''(x)$ are continuous and bounded on a certain interval containing $\alpha$. Let $f(\alpha) = 0$ , $f'(\alpha) = 0$, and $f''(\alpha) \neq 0$. Find the rate of convergence of the iterative method:
$x_{n+1} = x_{n} - 2\frac{f(x_{n})}{f'(x_{n})}$ which $n \geq 0 $.
The attempt: I have no idea how to start on this problem to be honest. Can someone give me a head start on this problem? Thank you very much!!