Answer
Diverges
Work Step by Step
Let us consider $a_n=(\dfrac{4n+3}{3n-5})^n$
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: $L=\lim\limits_{n \to \infty} \sqrt [n] {|a_n|}=\lim\limits_{n \to \infty} |a_n|^{1/n}$
$L=\lim\limits_{n \to \infty} \sqrt [n] {|a_n|}=\lim\limits_{n \to \infty}\sqrt [n] {(\dfrac{4n+3}{3n-5})^n}$
$\implies \lim\limits_{n \to \infty} \dfrac{4n+3}{3n-5}=\lim\limits_{n \to \infty} \dfrac{4+3/n}{3-5/n}$
and $L=\dfrac{4}{3} \gt 1$
Hence, the series Diverges by the Root Test.