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Test the given series for convergence or divergence

Determine if the following series converges or diverges and justify your decision.

    \[ \sum_{n=1}^{\infty} \frac{2^n n!}{n^n}. \]


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*}

Hence, the series converges.

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