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Test the given series for convergence or divergence

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

    \[ \sum_{n=1}^{\infty} \frac{(1000)^n}{n!}. \]


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{1000^{n+1}}{(n+1)!} \right) \left( \frac{n!}{1000^n} \right) \\[9pt]  &= \lim_{n \to \infty} \frac{1000}{n+1} \\[9pt]  &= 0 < 1.  \end{align*}

Hence, the series converges.

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