Answer
Diverges
Work Step by Step
Consider $a_n=\lim\limits_{n \to \infty} \dfrac{5 n}{4^n+3}$
But $\lim\limits_{n \to \infty} \dfrac{5 n}{4^n+3}=\dfrac{\infty}{\infty}$
Need to apply L-Hospital's rule.
$\lim\limits_{n \to \infty} \dfrac{ 5^n \ln 5}{4^n \ln 4} =(\frac{\ln 5}{\ln 4})\lim\limits_{n \to \infty} (\dfrac{5}{4})^n=\infty $
Hence, the given series diverges.
