Hi,

No, you made an algebraic error. Given $a_n=n3^{-n}$, then $a_{n+1}=(n+1)3^{-n-1}$ and so the quotient

$${a_n\over a_{n+1}}={n3^{-n}\over (n+1)3^{-n-1}}=3\,{n\over n+1}$$

So your interval of convergence is $|x-2|<3$ or $-1<x<5$.

Aside. When you find the interval of convergence of a power series, the ratio test or root test will always fail at the endpoints; the interval of convergence is found by one of these tests.