Answer
Diverges
Work Step by Step
Apply the ratio test to the series
$\Sigma^{\infty}_{n=1} \frac{5^n}{n^4}$
$\lim\limits_{n \to \infty} |\frac{\frac{5^{n+1}}{(n+1)^4}}{\frac{5^n}{n^4}}|$, Because all values are positive, absolute value is not needed.
$\lim\limits_{n \to \infty} \frac{5n^4}{(n+1)^4}$
Expand the denominator, then evaluate the limit by using the leading coefficients of the highest terms
$\lim\limits_{n \to \infty} \frac{5n^4}{n^4+4n^3 +6n^2 +4n+1} =5 \gt 1$
Because the answer is greater than 1, the series diverges
