Sorry, I won't analyze your solution and will just write what I think. According to this page, the radius of convergence is the reciprocal of . If , the limit is 1, so the radius is 1. If a > 1, the limit is 1 / a, so the radius is a.
Now, what happens when |x| equals the radius? If a < 1 and x = 1, the series diverges by the integral test. If a = x = 1, the series clearly diverges. If a > 1 and x = a, again the series diverges by the integral test. It is left to check when x equals negative radius.