Final answer:
The series ∑ (-1)^n/n^3 is absolutely convergent because the absolute series ∑ 1/n^3 is a convergent p-series with p > 1.the original series is absolutely convergent since the absolute value of the series converges
Step-by-step explanation:
To determine whether the given series is absolutely convergent, conditionally convergent, or divergent, we can use the Alternating Series Test. In this case, our series is (-1)^n * n^3 / 1
First, we need to check if the series satisfies the conditions of the Alternating Series Test:
The terms of the series approach zero as n approaches infinity. In this case, as n increases, n^3 grows faster than (-1)^n, meaning the terms of the series approach zero.
The terms of the series are decreasing. Since the absolute value of each term is given by n^3 / 1, which is an increasing function, the terms of the series are decreasing.
Since the given series satisfies both conditions of the Alternating Series Test, we can conclude that the series is conditionally convergent.
The student has asked to determine whether the series ∑ (-1)^n/n^3 from n = 1 to infinity is absolutely convergent, conditionally convergent, or divergent.
To solve this, we must look at the absolute convergence test. For a series to be absolutely convergent, the series formed by taking the absolute values of its terms must be convergent. Here, the absolute series would be ∑ 1/n^3. Since 1/n^3 is a p-series with p = 3 (> 1), this series converges by the p-series test. Therefore, the original series is absolutely convergent since the absolute value of the series converges.