Answer
Converges
Work Step by Step
Given
$$\sum_{n=1}^{\infty}\frac{n^3}{5^n}$$
By using Ratio Test, we get:
\begin{aligned} \rho &=\lim _{n \rightarrow \infty}\left|\frac{a_{n+1}}{a_{n}}\right| \\ &=\lim _{n \rightarrow \infty} \frac{(n+1)^{3}}{5^{n+1}} \cdot \frac{5^{n}}{n^{3}} \\ &=\lim _{n \rightarrow \infty} \frac{1}{5}\left(1+\frac{1}{n}\right)^{3} \\ &=\frac{1}{5} \end{aligned}
Thus the series converges.
