Answer
diverges
Work Step by Step
Here, we have $|a_n|=\Sigma_{n=1}^{\infty}(\dfrac{3}{2n+1})^{3n}$
Root Test states that when $\Sigma a_n$ is an infinite series with positive terms and, then $r=\lim\limits_{n \to \infty}\sqrt[n] |a_n|$
a) When $0 \leq r \lt 1$, the series converges. (b) When $r \gt 1$, or, $\infty$, so the series diverges. (c) When $r=1$, the ratio test is inconclusive.
Now, $r=\lim\limits_{n \to \infty}\sqrt[n] {(\dfrac{3}{2n+1})^{3n}}\\=\lim\limits_{n \to \infty} (\dfrac{3}{2n+1})^3\\=\lim\limits_{n \to \infty} (\dfrac{3}{2+1/n})^3\\=\dfrac{27}{8} \gt 1$
Therefore, the given series diverges by the root test.
