Use the Ratio Test.

$\lim_{n\to\infty}\Bigg|\frac{\frac{(n+1)\ln(n+1)}{3^{n+1}}}{\frac{n\ln(n)}{3^n}}\Bigg|$

$\lim_{n\to\infty}\Big|\frac{(n+1)\ln(n+1)}{3n\ln(n)}\Big|=?$

Compare this series to:

$\sum_{n=1}^\infty \frac{1}{n^{3/2}}$

This is an alternating series, so what is

$\lim_{n\to\infty}\frac{\ln(n)}{\ln(n^2)}=?$

$\ln(n^2)=2\ln(n)$

Use the Ratio Test.