Newton's method has a quadratic convergence under some conditions. However, I do not know how to show the quadratics convergence using an example.
To illustrate this, say $f(x) = \cos(x)- x^3$ and first guess $0.5$.
$n_1 = 1.112141637097$
$n_2 = 0.909672693736$
$n_3 = 0.867263818209$
$n_4 = 0.865477135298$
$n_5 = 0.865474033111$
But how can I show the rate of reduction of error when they all have the same number of digits？ how can I know which digits are accurate which are not? I am confused.