Answer
Diverges
Work Step by Step
Consider $a_n=(\dfrac{4n+3}{3n-5})^n$
By the Root Test $l=\lim\limits_{n \to \infty} \sqrt [n] {|a_n|}=\lim\limits_{n \to \infty} |a_n|^{1/n}$
$l=\lim\limits_{n \to \infty} \sqrt [n] {|a_n|}=\lim\limits_{n \to \infty}\sqrt [n] {(\dfrac{4n+3}{3n-5})^n}=\lim\limits_{n \to \infty} \dfrac{4n+3}{3n-5}$
or, $=\dfrac{4}{3}$
so, $l \gt 1$
Thus , the given series Diverges by the Root Test.
