I am confused about the definition of the rate of convergence.
The first definition I have seen at (1) says,
If a sequence $x_1, x_2, . . . , x_n$ converges to a value $r$ and if there exist real numbers $λ > 0$ and $α ≥ 1$ such that
$$\lim_{n\to\infty} \mid{\frac{x_{n+1} - r}{(x_n-r)^\alpha}}\mid = \lambda $$ Then $\alpha$ is the rate of convergence of the sequence.
The second is from the wikipedia page on rate of convergence.
If a sequence $x_1, x_2, . . . , x_n$ converges to a value $r$ and if there exist real numbers $\mu \in (0,1)$ such that
$$\lim_{n\to\infty} \mid{\frac{x_{n+1} - r}{x_n-r}}\mid = \mu $$
Then $x_n$ converges linearly at rate $\mu$.
The confusion arises when suppose in the first definition for $\alpha = 1$ and $\lambda = 1/2$ the rate of convergence would be $1$. In the second definition my rate of convergence would be $1/2$. What have I missed?
The reason for asking is that I am trying to find the rate of convergence of a sequence $x_n$ which has limit $0$.
$$\lim_{n\to\infty} \mid{\frac{x_{n+1}}{x_n}}\mid = \lambda >0 $$
Where $\lambda \in (0,\infty)$ depending on parameters of $x_n$. The first definition would say $x_n$ converges at rate 1. The second would suggest the rate will depend on my parameters for $x_n$.
(1) http://www.math-cs.gordon.edu/courses/ma342/handouts/rate.pdf
The English Wikipedia page you cite uses the phrase "converges with order $\alpha$ to $r$" for your first display equation and uses the phrase "converges linearly to $r$" with "rate of convergence $\mu$" for your second display equation. This reference does not use the phrase "rate of convergence" for the first form you give.
Using these terms, the "rate of convergence" is only defined if convergence is of order $1$.
The paper you cite uses the phrase "rate of convergence" to replace the phrase "converges with order" as it is used on the Wikipedia page. it is not unusual for papers to have nonstandard uses of phrases. It strongly suggests that if you are going to use these terms, you should lay out explicitly what they mean in your own speaking and writing.