Consider the recurrence relation $x_n=2(x_{n-1}+x_{n-2})$ and the general solution $z_n=\alpha(1+\sqrt{3})^n+\beta(1-\sqrt{3})^n$.
For which values of $\alpha,\beta\in\mathbb{R}$ does the general solution converge? Determine the order of the rate of convergence for these values.
I've found $z_n$ to be convergent for $\alpha=0$ and $\beta\in (-\infty,\infty)$.
But how do I go about determining the the order of the rate of convergence?
When there is convergence,
$$z_n=\beta(1-\sqrt3)^n$$ is simply a geometric progression of common ratio $1-\sqrt3$ and the convergence is linear.