Show how you could use the Newton-Raphson method on $f(x) = x^3 - a$ to calculate $a^{1/3}$. Does it converge? Prove your answer.
This is what I have:
If $x^*$ is a root of $f(x)$, then $f(x^*) = 0 \Rightarrow (x^*)^3 -a = 0 \Rightarrow x^* = a^{1/3}$. Thus $x^* = a^{1/3}$ is the only root of the function $f(x)$.
To calculate the root $a^{1/3}$ we can use Newton-Raphson,
$\displaystyle{x_{n+1} = x_{n} -\frac{(x_n)^3 -a}{3(x_n)^2}}$
How can I prove that this converges or fails to converge?