We can apply the ratio test either to the original series, which is an expansion around $x=1/3$, or we can apply it to the transformed series by taking $z = 3x-1$.
If we do the former, the ratio test gives a radius of convergence of $1/6$ around $x=1/3$. If we do the latter, the radius of convergence is $1/2$ around $z=0$. So either $x \in (1/6,1/2)$ or $z \in (-1/2,+1/2)$ will be an interval of absolute convergence, according to the ratio test.
Now we get an alternating series at the endpoint $x=1/6$, equiv. $z=-1/2$, and because of the $n$ in the denominator, this will be a conditionally convergent series (harmonic) we are familiar with.
At the other endpoint the series is all positive terms/harmonic, and we know that diverges (very, very slowly).