I have the equation $f(x)=1/x-a=0$ and I need to find necessary and sufficient conditions for the start value $x_0$ so that the method converges.
First I set up the sequence $x_{k+1}=x_k-\frac{f(x_k)}{f'(x_k)}=-ax_k^2+2x_k$ (Is this correct?) Now for convergence I thought that $\frac{||x_{k+1}-\frac{1}{a}||}{||x_k-\frac{1}{a}||}$ has to be less than 1 (1/a in this case is the exact solution of the equation). Is this true? And if so, how would I show that? Also how could I then derive conditions for the initial value $x_0$?