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Determine the convergence of the series (-1)n / (n log2 (n+1))

Consider the series

    \[ \sum_{n=1}^{\infty} \frac{(-1)^n}{n \log^2 (n+1)}. \]

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 can see this since

    \[ \sum_{n=1}^{\infty} \left| \frac{(-1)^n}{n \log^2 (n+1)} = \sum_{n=1}^{\infty} \frac{1}{n \log^2 (n+1)}. \]

This series converges by the integral test (example #2 on page 398 of Apostol). \qquad \blacksquare

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