Question

Which of the following series is absolutely convergent?

I. ∑(n=1 to [infinity]) (-1)^(n+1) / ∛(n⁴)
II. ∑(n=1 to [infinity]) (-1)^n / n!
III. (Please provide the third series here)

Answer 1

The first series is conditionally convergent, while the second series is absolutely convergent.

Step-by-step explanation:

The series that are absolutely convergent are those that converge regardless of the order in which the terms are summed. In order to determine if a series is absolutely convergent, we need to look at the absolute values of the terms. If the series of absolute values converges, then the original series is absolutely convergent. If not, then it is conditionally convergent or divergent.

Let's analyze each series:

I. The series ∑(n=1 to [infinity]) (-1)^(n+1) / ∛(n⁴) is conditionally convergent. We can see that the terms alternate in sign, which suggests the use of the Alternating Series Test to determine convergence. However, the series of absolute values, ∑(n=1 to [infinity]) |(-1)^(n+1) / ∛(n⁴)|, diverges.

II. The series ∑(n=1 to [infinity]) (-1)^n / n! is absolutely convergent. This is because the terms approach zero as n tends to infinity, and the series of absolute values, ∑(n=1 to [infinity]) |(-1)^n / n!|, converges.

III. It seems that the third series is missing in the question. Please provide the third series, and I will be happy to analyze it for you.