Determine if the following series converges or diverges and justify your decision.
![\[ \sum_{n=1}^{\infty} \frac{3^n n!}{n^n}. \]](https://www.stumblingrobot.com/wp-content/ql-cache/quicklatex.com-747bd15b00313f28587e30510fdde39e_l3.png "Rendered by QuickLaTeX.com")
This series diverges by the ratio test since
![\begin{align*} \lim_{n \to \infty} \frac{a_{n+1}}{a_n} &= \lim_{n \to \infty} \left( \frac{3^{n+1}(n+1)!}{(n+1)^{n+1}} \right) \left( \frac{n^n}{3^n n!} \right) \\[9pt] &= 3 \lim_{n \to \infty} \left( \frac{n}{n+1}\right)^n \\[9pt] &= 3 \lim_{n \to \infty} \left( \frac{1}{1+\frac{1}{n}} \right)^n \\[9pt] &= \frac{3}{e} > 1. \end{align*}](https://www.stumblingrobot.com/wp-content/ql-cache/quicklatex.com-88090ee1ce515cf60a065d34db80c477_l3.png "Rendered by QuickLaTeX.com")
Hence, the series diverges.
Determine if the following series converges or diverges and justify your decision.
This series diverges by the ratio test since
Hence, the series diverges.