Answer
the series $\sum_{n=1}^{\infty} \frac{ 1}{3n^4+12n}$ converges
Work Step by Step
Use the limit comparison test with the convergent p-series $\sum_{n=1}^{\infty} \frac{1}{n^4}$
Since we have
$$L=\lim_{n\to \infty } \frac{1/(3n^4+12n)}{1/n^4}=\lim_{n\to \infty } \frac{ n^4}{3n^4+12n}\\
=\lim_{n\to \infty } \frac{ 1}{3+12/n^3}=1/3\gt0$$
then the series $\sum_{n=1}^{\infty} \frac{ 1}{3n^4+12n}$ converges because the p-series $\sum_{n=1}^{\infty} \frac{ 1}{ n^4 }$, $p=4\gt1$, converges.
