Let $f(x) = (x-r_1)(x-r_2)...(x-r_d)$ where $r_1 < r_2 < ... < r_d$. I need to prove that if $x_0 > r_d$, Newton-Raphson converges to $r_d$.
I could see that:
$$f(x) = \prod_{j=1}^d (x-r_j) $$ $$\frac{d}{dx}f(x) = \sum_{i=1}^d \prod_{i \neq j}(x-r_j)$$
But I´m stuck.