quadratic convergence

A sequence $\{x_i\}$ in a metric space $(X,d)$ is said to converge quadratically to $x^{*}$ if there is a constant $1>c>0$ such that $d(x_{i+1},x^{*})\le cd{(x_i,x^{*})}^2$ for all $i$.

The convergence is said to be of order $p$ if $d(x_{i+1},x^{*})\le cd{(x_i,x^{*})}^p$ for all $i$.