Answer
Diverges
Work Step by Step
Given: $a_{n+1}=(\dfrac{3n-1}{2n+5})a_n$ and $a_1=\dfrac{1}{3}$
$\lim\limits_{n \to \infty} |\dfrac{a_{n+1}}{a_{n}} |=\lim\limits_{n \to \infty}|(\dfrac{\dfrac{3n-1}{2n+5})a_n}{a_n}|$
$\lim\limits_{n \to \infty}|\dfrac{3n-1}{2n+5}|=\lim\limits_{n \to \infty}|\dfrac{3-\dfrac{1}{n}}{2+\dfrac{5}{n}}|=\dfrac{3-0}{2+0}=\dfrac{3}{2} \gt 1$
Therefore, the series diverges by the ratio test.