Answer
Divergent
Work Step by Step
Let us consider $f_n=\lim\limits_{n \to \infty} \dfrac{5 n}{4^n+3}$
Here, $\lim\limits_{n \to \infty} \dfrac{5 n}{(4^n+3)}=\dfrac{\infty}{\infty}$
This shows that the limit has an indeterminate form so we will use L-Hospital's rule.
$\lim\limits_{n \to \infty} \dfrac{ (5^n) \ln 5}{(4^n) \ln 4} =\dfrac{\ln 5}{\ln 4} \cdot [\lim\limits_{n \to \infty} (\dfrac{5}{4})^n]=\infty $
Thus, the series is divergent.