Answer
The series $\Sigma_{n=1}^{\infty} \frac{n^5}{5^n}$ converges.
Work Step by Step
To apply the ratio test, we have:
$$
\rho=\lim _{n \rightarrow \infty}\left|\frac{a_{n+1}}{a_{n}}\right|=\lim _{n \rightarrow \infty} \frac{(n+1)^5/ (5)5^n}{n^5/5^n}=\frac{1}{5}\lim _{n \rightarrow \infty} \frac{(n+1)^5}{n^5}=\frac{1}{5}\lt 1
$$
Hence, the series $\Sigma_{n=1}^{\infty} \frac{n^5}{5^n}$ converges.
