Determine if the following series converges or diverges and justify your decision.
![\[ \sum_{n=1}^{\infty} \frac{2^n n!}{n^n}. \]](https://www.stumblingrobot.com/wp-content/ql-cache/quicklatex.com-fffed4f12a172a9a62fdec691e1e1a49_l3.png "Rendered by QuickLaTeX.com")
The series converges by the ratio test since
![\begin{align*} \lim_{n \to \infty} \frac{a_{n+1}}{a_n} &= \lim_{n \to \infty} \left( \frac{2^{n+1}(n+1)!}{(n+1)^{n+1}} \right) \left( \frac{n^n}{2^n n!} \right) \\[9pt] &= \lim_{n \to \infty} \frac{2(n+1)n^n}{(n+1)^{n+1}} \\[9pt] &= \lim_{n \to \infty} \frac{2n^n}{(n+1)^n} \\[9pt] &= 2 \lim_{n \to \infty} \left( \frac{n}{n+1} \right)^n \\[9pt] &= 2 \lim_{n \to \infty} \left( \frac{1}{1+\frac{1}{n}} \right)^n \\[9pt] &= \frac{2}{e} < 1. \end{align*}](https://www.stumblingrobot.com/wp-content/ql-cache/quicklatex.com-1b77adeb6f63ba2bac9b5b5eec7f9eaf_l3.png "Rendered by QuickLaTeX.com")
Hence, the series converges.
Determine if the following series converges or diverges and justify your decision.
The series converges by the ratio test since
Hence, the series converges.