Hi I am trying to do a question on rate of convergence
I have come across the definition
If a sequence $(x_n)$ converges to a value $r$ and if there exist real numbers $\beta > 0$ and $\alpha \geq 1$ such that
$$\lim_{n\to \infty} \frac{|x_{n+1} - r|}{|x_n - r|} $$
then $\alpha$ is the rate of convergence.
I am trying to obtain alpha for the following question or rather obtain alpha using the above expression.
$$a_n = \frac{(-1)^n}{6^n-4}$$
Therefore from the given expression i know i need to obtain $a_{n+1}$ where
$$a_{n+1} = \frac{(-1)^{n+1}}{6^{n+1}-4}$$
$$\lim_{n\to \infty} \frac{|\frac{(-1)^{n+1}}{6^{n+1}-4} |}{|\frac{(-1)^n}{6^n-4}|}= $$
$$\lim_{n\to \infty} \frac{6^n-4}{6^{n+1}-4} = \frac{1}{6} $$
Evaluating the above limit gives $\frac{1}{6}$