Answer
Converges
Work Step by Step
Consider $a_n=\dfrac{4^n}{(3n)^n}$
By the Root Test $l=\lim\limits_{n \to \infty} \sqrt [n] {|a_n|}=\lim\limits_{n \to \infty} |a_n|^{1/n}$
$l=\lim\limits_{n \to \infty} \sqrt [n] {|a_n|}=\lim\limits_{n \to \infty}|\sqrt [n] {\dfrac{4^n}{(3n)^n}}|=\lim\limits_{n \to \infty} \dfrac{4}{(3n)}$
or, $=\dfrac{1}{\infty}$
so, $l =0\lt 1$
Thus , the given series converges by the Root Test.