Consider the series
![\[ \sum_{n=1}^{\infty} (-1)^n \left( \frac{1 \cdot 3 \cdot 5 \cdots (2n-1)}{2 \cdot 4 \cdot 6 \cdots (2n)} \right)^3. \]](https://www.stumblingrobot.com/wp-content/ql-cache/quicklatex.com-79b850f53864e60612954f8817061616_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
![\[ \sum_{n=1}^{\infty} \left| (-1)^n \left(\frac{1 \cdot 3 \cdot 5 \cdots (2n-1)}{2 \cdot 4 \cdot 6 \cdots (2n)}\right)^3 \right| = \sum_{n=1}^{\infty} \left( \frac{1 \cdot 3 \cdot 5 \cdots (2n-1)}{2 \cdot 4 \cdot 6 \cdots (2n)} \right)^3. \]](https://www.stumblingrobot.com/wp-content/ql-cache/quicklatex.com-29261cce389bc7f6878d37eef23c39b9_l3.png "Rendered by QuickLaTeX.com")
This series converges by a previous exercise (Section 10.16, Exercise #18) with 