newton raphson method convergence problem

Question

My problem is:

An iterative method to find $n$-th root of a positive number $a$ is given by $x_{k+1}=\frac{1}{2} \left[x_k +\frac{a}{x_k^{n-1}}\right]$

Find the value of $n$ for which this iterative method fails to converge.

I tried to use $|g'(x)|<1$ but could not get it .

Please help

Answer 1

I hope you found that for $$g(x)=\frac12(x+a/x^{n-1})$$ the derivative is $$g'(x)=\frac12(1-(n-1)a/x^n).$$

To have a useful numerical method this needs to be contractive at least in the solution of the problem. There $$ g'(\sqrt[n]a)=1-n/2 $$ which has to fall inside the interval $(-1,1)$.