Questions about fixed point iteration

I am learning fixed-point iteration and am confused about the convergence rate, which is defined as follows:

$$\lim_{k \rightarrow\infty}\frac{x_{k+1}-x^*}{(x_k-x^*)^p}=C,\quad C\neq 0$$ Then we call the iteration process is $p$-order convergent.

My question is when the iteration process is $p+1$ order convergent, is it $p, p-1, \cdots, 1$ order convergent?

My thoughts, suppose we have

$$\lim_{k \rightarrow\infty}\frac{x_{k+1}-x^*}{(x_k-x^*)^{p+1}}=C,\quad C\neq 0$$

then, $$\lim_{k \rightarrow\infty}\frac{x_{k+1}-x^*}{(x_k-x^*)^{p}}*\frac{1}{(x_k-x^*)}=C,\quad C\neq 0$$

If the first term converges to some constant, then contradicts coz the second term goes to infinity.

Solution 1:

If a sequence $(x_k)$ converges to $x^*$ and we have that, for some $p$, $$ \lim_{k\to \infty} \dfrac{|x_{k+1}-x^*|}{|x_k-x^*|^p} = C_p \ne 0 $$

then,

• for any $q < p$, we have that $$ \lim_{k\to \infty} \dfrac{|x_{k+1}-x^*|}{|x_k-x^*|^q} = 0 $$

• for any $q>p$, we have that $$ \lim_{k\to \infty} \dfrac{|x_{k+1}-x^*|}{|x_k-x^*|^q} = \infty $$

So, you see that the order of convergence, if finite, is uniquely determined.