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 \(\boxed{\text{Absolutely convergent}}\).