The alternating series test says that this converges. (It happens to converge to , but that's neither here nor there.)
2. You don't need an -proof. I would try to prove that it's monotonically decreasing, and convergence would follow from there (because it's obviously bounded below by ). You should be able to do it by induction, though I won't go into the details, because it's likely to be tedious. Once you know that it converges, finding the number to which it converges is easy:
According to the equation, is also a solution, but we can ignore it because is bounded below by .