What is the order of convergence of Newtons root finding method? And when does it converge?

Question

Given a function $f(x)$, we can approximate $x_r$ where $f(x_r)=0$ , by using Newton's method:

$$x_{n+1}=x_n -\frac{f(x_n)}{f'(x_n)} $$

The method only works when 'you choose an $x_0$ near enough to $x_r$'.

My questions are:

1. When does this series converge?

2. Given that the series converges, what is the order of convergence?

If it depends on $f(x)$, then let me define $f(x)=e^x-x-1.9\cos{x}$

Answer 1

When $|f'(x)|<1 \ \forall x \in (a,b), \ f(x)$ will converges to a root in $(a,b)$ given a start value in $(a,b)$. The rate of convergence is quadratic for roots of multiplicity $m=1$. for $m>1$ convergence is linear but in that case $x_{n+1}=x_n-m\frac{f(x_n)}{f'(x_n)}$ will have a quadratic convergence rate.