Answer
Diverges
Work Step by Step
Let us consider $a_n=\dfrac{3^{n+2}}{ \ln n}$
In order to solve the given series we will take the help of Ratio Test. This test states that when the limit $L \lt 1$ , the series converges and for $L \gt 1$, the series diverges.
$L=\lim\limits_{n \to \infty} |\dfrac{a_{n+1}}{a_{n}} |=\lim\limits_{n \to \infty}|\dfrac{\dfrac{3^{n+1+2}}{ \ln (n+1)}}{\dfrac{3^{n+2}}{ \ln n}}|$
$\implies \lim\limits_{n \to \infty}|\dfrac{3 \ln n}{\ln (n
+1)}|$
and $L=3 \gt 1$
Hence, the series Diverges absolutely by the ratio test.