#### Answer

Converges

#### Work Step by Step

Let us consider $a_n=\dfrac{4^n}{(3n)^n}$
In order to solve the given series we will take the help of Root Test. This test states that when the limit $L \lt 1$, the series converges and for $L \gt 1$, the series diverges .In order to solve the series we will take the help of Root Test.
It can be defined as follows: $L=\lim\limits_{n \to \infty} \sqrt [n] {|a_n|}=\lim\limits_{n \to \infty} |a_n|^{1/n}$
$L=\lim\limits_{n \to \infty} \sqrt [n] {|a_n|}=\lim\limits_{n \to \infty}|\sqrt [n] {\dfrac{4^n}{(3n)^n}}|$
$\implies \lim\limits_{n \to \infty} \dfrac{4}{(3n)}=\dfrac{1}{\infty} =0\lt 1$
Hence, the series converges by the Root Test.