It's my homework assigment and I don't expect written ready proof, but would like some guidance and hints about how to go on doing this.
It is asked to prove that Euler-Chebyshov and Halley's iteration methods are cubic convergent, and for solution holds :$f(x) = 0$ and $f'(x) \neq 0$.
Euler-Ch. method:
$x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)} - \frac{f''(x_n)}{2f'(x_n)} \cdot \left( \frac{f(x_n)}{f'(x_n) } \right)^2$
Halley's method:
$x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n) - \frac{1}{2}\cdot f''(x_n) \cdot \frac{f(x_n)}{f'(x_n)}}$
I know that for cubic convergence holds $\mathcal{E} = |x_n - x_s|$, $\mathcal{E}_{n+1} = 10^{-3}, \mathcal{E}_{n+2} = 10^{-9} . . .$ Where $x_s$ is solution.
How to begin the proof ?