I'm working on a homework problem about Halley's method and I'm not quite sure where to start and how I prove a cubic convergence.
Consider the iterative method $$x_{n+1} = x_{n} − \frac{h_n}{Ah_n + B},\space\space hn := \frac{f(x_n)}{f'(x_n)}$$
Determine A and B (depending of $f$ and its derivatives at $x$ = $x_k$) such that the method has cubic order of convergence.
Compare with the Taylor expansion of $$ f(x_n-(h_n+v))=-f'(x_n)v+\frac12f''(x_n)(h_n+v)^2+O((h_n+v)^3) $$ where $v=O(f(x)^2)$ or $v=O((h_n)^2)$, and write the given expression as $$ x_{n+1}=x_n-\frac{h_n}{B}+\frac{Ah_n^2}{B(B+Ah_n)} $$ which should with relatively little effort show that $B=1$ and then also give the value of $A$ that makes the terms up to the quadratic cancel out.