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Test the given series for convergence or divergence

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

    \[ \sum_{n=1}^{\infty} \frac{n!}{3^n}. \]


The 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{(n+1)!}{3^{n+1}} \right) \left( \frac{3^n}{n!} \right) \\[9pt]  &= \lim_{n \to \infty} \frac{n+1}{3} \\[9pt]  &= +\infty > 1. \end{align*}

Hence, the series diverges.

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