Cfrgtkky

By definition, a sequence $x_k in mathbb{R}, k in mathbb{N}$
converges with order $p in [1,infty)$ to $x := lim_{ktoinfty}
x_k$ if begin{align} exists C in [0,infty): forall k in
mathbb{N}: |x_{k+1} - x| le C |x_k - x|^p end{align}

Assume $x_k$ is a sequence of approximations to $x$, then $|x_k - x|$ denotes the error of approximation at $k$-th iteration, and goes to zero as $k to infty$. I can understand the inequality $|x_{k+1} - x| le C |x_k - x|$ that implies the error must go smaller and smaller along the iterations. But I don't understand the motivation to put a power of $p$ on the previous step. Any practical meaning of taking $p$-th power of error $|x_k - x|^p$?

Consider a sequence of positive numbers $x_n$ such that $x_{n+1} = x_n^p$ with $p > 1$ and $x_0 < 1$. That's a special case, where $C = 1$.

Now take the logarithm. You should then be able to find an explicit formula for $x_n$. As $p$ varies, how does the rate change at which $x_n$ converges to 0? For example, how many steps are needed to guarantee $x_n < varepsilon$ for a given $varepsilon$?

By the way, the estimate $|x_{k+1} - x| le C |x_k - x|$ does not imply that the error goes to zero. This happens only if $C < 1$.