Answer
Diverges
Work Step by Step
$$\lim _{n\rightarrow \infty }\dfrac {3^{n}}{n^{3}}=\dfrac {\dfrac {d}{dn}\left( 3^{n}\right) }{\dfrac {d}{dx}\left( n3\right) }=\dfrac {3^{n}\ln 3}{3n^{2}}=\dfrac {\dfrac {d}{dn}\left( 3^{n}\ln 3\right) }{\dfrac {d}{dn}\left( 3n^{2}\right) }=\dfrac {3^{n}\left( \ln 3\right) ^{2}}{6n}=\dfrac {\dfrac {d}{dn}\left( 3^{n}\left( \ln 3\right) ^{2}\right) }{\dfrac {d}{dn}\left( 6n\right) }=\dfrac {3^{n}\left( \ln 3\right) ^{3}}{6}=\infty \neq 0$$
$$\Rightarrow \sum ^{\infty }_{n=1}\dfrac {3^{4}}{n^{3}}$$ diverges