Answer
Divergent
Work Step by Step
$\Sigma_{n=1}^{\infty}\frac{5^{k}}{3^{k}+4^{k}}\gt \Sigma_{n=1}^{\infty}\frac{5^{k}}{4^{k}+4^{k}}$
$=\Sigma_{n=1}^{\infty}\frac{5}{8} (\frac{5}{4})^{k-1}$
This is a geometric series with a common ratio $r=\frac{5}{2}$, hence diverging.