Why does this method for finding cube roots work? While on the Internet I came across a formula for cube root using recursion. The formula was:
$$ use \space x_1 = \frac{a}{3}  \space (a \space is \space the \space number \space we \space want \space to \space find \space cube \space root \space of)$$
Then use the formula:
$$ x_{n+1} = \frac 13 \left( 2x_n + \frac{a}{{x_n}^2} \right)$$
Recursively.
What is this method and how does it work. Is their someway I can prove it ?

Suppose that $a \ne 0.$ We have that 
$$
x^3=a \iff x =\frac 13 \left( 2x+\frac a{x^2} \right).
$$
Thus the problem is reduced to finding the fixed point of the function 
$$
f(x)=\frac 13 \left( 2x+\frac a{x^2} \right),
$$
which can be done with the use of the method of fixed-point iterations. Google the phrase "method of fixed-point iterations".
