I haven't found a proof of convergence yet, but the limit (assuming it converges) is
the sequence has a stable point of equilibrium when
Solving this for gives
Since the square root of this sequence is the sequence we want to find the limit of, these are the equilibrium points of our sequence. Since it is readily apparent that we initially move toward , we can tell that if the sequence converges, it will converge to this point.
I will post again if I find a proof it converges.