My book claims that a serie has a quadractic convergence if $\forall x_k$, $k = 1, 2, \dots$: \begin{equation} |x-x_{k+1}| \le C|x-x_k|^p \end{equation} such as $C \in \mathbb{R}$ and $p = \color{red}2$.
However, it noticed it wasn't that simple: the Newton-Raphson method (used to find roots of a real function $f \in C^1(I \subseteq \mathbb{R}, \mathbb{R})$) for instance which has a quadratic convergence, diverges if the root you are looking for is a double root ($f(x_0) = 0$ such as $f'(x_0) = 0)$.
Here is my question: What are the complete hypotheses of the quadratic convergence of a real serie $x_k$ ?