TITLE: When does Newton-Raphson Converge/Diverge?
QUESTION [7 upvotes]: Is there an analytical way to know an interval where all points when used in Newton-Raphson will converge/diverge? 
I am aware that Newton-Raphson is a special case of fixed point iteration, where:
$$ g(x) = x - \frac{f(x)}{f'(x)} $$
Also I've read that if $|f(x)\cdot f''(x)|/|f'(x)^2| \lt 1$, then convergence is assured. I am just not sure on how to use this fact? Could anyone give me some examples? Thanks.

REPLY [3 votes]: A theoretically nice but practically nearly useless answer is provided by the Newton-Kantorovich theorem: If $L=M_2$ is an upper bound for the magnitude of the second derivative over some interval $I$, and with $x_0\in I$ and the first step $s_0=-\frac{f(x_0)}{f'(x_0)}$ the "ball" $B(x_0+s_0,|s_0|)=(x_0+s_0-|s_0|,x_0+s_0+|s_0|)$ is contained in $I$ and
$$
L·|f'(x_0)^{-1}|^2·|f(x_0)|\le\frac12
$$
then there is a unique root inside that ball and Newton's method converges towards it.