Answer
Diverges
Work Step by Step
Let us suppose that the function $f(x)=\dfrac{3x^2+1}{4x^3+2}$
This implies that $f'(x)=\dfrac{-12(x^4+x^2-x)}{(4x^3+2)^2}$
Next, $\lim\limits_{n \to \infty} n a_n=n \times \lim\limits_{n \to \infty} \dfrac{3n^2+1}{4n^3+2}=\dfrac{3}{4}$
and $\dfrac{3}{4} \gt 0$
This means that the given series has a finite value, so it is diverges.