# Convergence of Newton Raphson when derivative at root is 0

How would one show that the newton raphson method converges linearly in these 2 cases?

$i) f'(\alpha)=0, f''(\alpha)\ne0$
$ii)f'(\alpha)=0, f''(\alpha)=0, f'''(\alpha)\ne0$ where $\alpha$ is the root of the function. The hint I have been given is to use L'Hopitals rule, but cant quite figure it out.

• – Lutz Lehmann May 12 '17 at 16:17

Write $f(x)=(x-α)^m·g(x)$ with $g(α)\ne 0$ and use $x\approx α$ to find $$x-\frac{f(x)}{f'(x)}\approx α+\frac{m-1}m(x-α)$$