1. Is this series absolutely convergent, conditionally convergent, or divergent?
\[
\sum_{n=0}^{\infty} \frac{12(-1)^{n+1}}{\sqrt[3]{n^{6}+5 n^{4}+8}}
\]
Absolutely convergent
Conditionally convergent
Divergent

Step 1 ：We are given the series \(\sum_{n=0}^{\infty} \frac{12(-1)^{n+1}}{\sqrt[3]{n^{6}+5 n^{4}+8}}\). We need to determine if this series is absolutely convergent, conditionally convergent, or divergent.

Step 2 ：First, we note that this is an alternating series. To determine its convergence, we can apply the Alternating Series Test and the Absolute Convergence Test.

Step 3 ：The Alternating Series Test states that if the absolute value of the terms in the series are decreasing and the limit as n approaches infinity is zero, then the series is conditionally convergent.

Step 4 ：Let's apply the Alternating Series Test. We find that the derivative of the term is negative, which means the absolute value of the terms are decreasing. Also, the limit as n approaches infinity is zero. Therefore, the series is conditionally convergent according to the Alternating Series Test.

Step 5 ：Next, we apply the Absolute Convergence Test. This test states that if the series of the absolute value of the terms is convergent, then the series is absolutely convergent.

Step 6 ：We find that the series of the absolute value of the terms is convergent. Therefore, the original series is absolutely convergent according to the Absolute Convergence Test.

Step 7 ：Final Answer: The series is \(\boxed{\text{Absolutely convergent}}\).