Looks good so far.

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.

gives

so

.

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...