Consider the following text taken from this link.
What does it say?
As far as I can get:
If $p_n$ (i.e. $p_0, p_1, p_2, ...$) is a sequence, $p_n$'s point of convergence is $p$, $\lambda$ and $\alpha$ (where $\alpha < 0$) are constants, and $$\lim_{n \rightarrow \infty} \frac{\left| p_{n+1} - p \right|}{\left|p_n - p\right|} = \lambda $$ (i.e. if $n$ goes to infinity, then, the ratio of "difference of the $n$-th $p$ and $p$", and the "$n+1$-th $p$ and $p$" would be $\lambda$) [$\text{What does it mean though?}$],
then we can say that Rate of Convergence is ... ????
Can anyone help me to complete the rest of the text in pain English?
You've got the $\alpha$ inequality wrong (it's $\alpha>0$) and you're missing an $\alpha$ in the denominator, that should read $|p_n-p|^\alpha$. Perhaps it will help you to write
$$\lim_{n \to \infty}\frac{|p_{n+1}-p|}{|p_{n}-p|^\alpha}=\lim_{n \to \infty}\frac{|\frac{p_{n+1}}{p}-1|}{|\frac{p_n}{p}-1|^\alpha}.$$
To me, this means that the rate of change of the series tends to stabilize as $n$ grows.