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Determine the convergence of the series (-1)n n37 / (n+1)!

Consider the series

    \[ \sum_{n=1}^{\infty} \frac{(-1)^n n^{37}}{(n+1)!}. \]

Determine whether the series converges or diverges. If it converges, determine whether it converges conditionally or absolutely.


The given series is absolutely convergent.

Proof. We have

    \[ \sum_{n=1}^{\infty} \left| \frac{(-1)^n n^{37}}{(n+1)!} \right| = \sum_{n=1}^{\infty} \frac{n^{37}}{(n+1)!}. \]

This sum converges 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)^{37}}{(n+2)!} \right) \left( \frac{(n+1)!}{n^{37}} \right) \\[9pt]  &= \lim_{n \to \infty} \left( \frac{1}{n+2} \cdot \frac{(n+1)^{37}}{n^{37}} \right) \\[9pt]  &= 0 < 1. \end{align*}

By the ratio test the series converges. Hence, the given series converges absolutely. \qquad \blacksquare

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