Answer
Diverges
Work Step by Step
We are given that $a_{n+1}=\dfrac{3n-1}{2n+5}a_n$ and $a_1=\dfrac{1}{3}$
Now, $l=\lim\limits_{n \to \infty} |\dfrac{a_{n+1}}{a_{n}} |=\lim\limits_{n \to \infty}|\dfrac{\dfrac{3n-1}{2n+5}a_n}{a_n}|$
Thus, we have $l=\lim\limits_{n \to \infty}|\dfrac{3n-1}{2n+5}|=\lim\limits_{n \to \infty}|\dfrac{3-1/n}{2+5/n}|$
So, $l=\dfrac{3}{2} \gt 1$
Hence, the series Diverges by the ratio test.