Consider the series
![\[ \sum_{n=1}^{\infty} \frac{(-1)^n}{n \log^2 (n+1)}. \]](https://www.stumblingrobot.com/wp-content/ql-cache/quicklatex.com-f3cbc50d3df2ff6f533d57d41ad4cbd7_l3.png "Rendered by QuickLaTeX.com")
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
![\[ \sum_{n=1}^{\infty} \left| \frac{(-1)^n}{n \log^2 (n+1)} = \sum_{n=1}^{\infty} \frac{1}{n \log^2 (n+1)}. \]](https://www.stumblingrobot.com/wp-content/ql-cache/quicklatex.com-fc341930cfbac9b734f562c5996c94f7_l3.png "Rendered by QuickLaTeX.com")
This series converges by the integral test (example #2 on page 398 of Apostol)
“This series converges by the integral test (example #2 on page 398 of Apostol)” That example is different but you can prove it by comparison.