I don't see your point here. What is important is the limit as n goes to infinity. What happens when n= 1 is irrelevant.Starting point n = 1
where I have divided both numerator and denominator by n. As n goes to infinity, both of those fractions, with n in the denominator, go to 0. The limit is . Also, everything should be in absolute value signs. In the limit, you have and, by the "ratio test" that must be less than 1.
That means that the radius of convergence is and the series converges (uniformly) inside the circle centered at with radius .
I'd really appreciate if someone pointed me in the right direction...