Determine values for which the general solution converges Textbook problem.  

Given the following general solution to a recurrence relation
  $$z_n = \alpha(1+\sqrt{3})^n + \beta(1-\sqrt{3})^n$$
  For which values $\alpha, \beta$ does the solution converge? And determine the order of the rate of convergence for these values.

By attempting to plot the sequence in some interval with varying values of $\alpha, \beta$ it seems like it will converge whenever $\alpha=0$ and $\beta = (-\infty, \infty)$, but how can i go about determining this in a more rigorous way?

You are correct.
For $a \neq 0$ we have that $z_n \to +\infty$ or $-\infty$  since $(1+\sqrt{3})>1$
Also $(1-\sqrt{3})<1 $
Thus the sequence converges $\forall b \in \Bbb{R}$ and for $a=0$
And converges to zero ,for every such value.
