Answer
Diverges
Work Step by Step
Here, we have $a_n=(\dfrac{3n-1}{2n+5})^n$
Root Test states that when $\Sigma a_n$ is an infinite series with positive terms and, then $l=\lim\limits_{n \to \infty}\sqrt[n] a_n$
a) When $0 \leq l \lt 1$, the series converges. (b) When $l \gt 1$, or, $\infty$, so the series diverges. (c) When $l=1$, the ratio test is said to be inconclusive.
Now, $l=\lim\limits_{n \to \infty} |\sqrt[n] {(\dfrac{3n-1}{2n+5})^n}| \\=\lim\limits_{n \to \infty} |\dfrac{3n-1}{2n+5}|=\dfrac{3}{2} \gt 1$
Therefore, the series diverges by the root test.
