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?

Since $z_n$ coverges it is bounded, so the sequence $${z_n\over (1+\sqrt{3})^n}$$ also coverges and it limit is $0$, so:
$$ 0 = \alpha +\beta \lim_{n\to \infty}{(1-\sqrt{3})^n\over (1+\sqrt{3})^n}= \alpha+\beta\cdot 0\implies \alpha =0$$
and clearly $\beta $ can be any real number.
