Question

Which of the following series is absolutely convergent?

I. Σ[n=1 to [infinity]] ((-1)^(n+1)) / n^(4/3)
II. Σ[n=1 to [infinity]] ((-1)^n) / n!
III. Σ[n=1 to [infinity]]

Answer 1

The series Σ[n=1 to ∞] ((-1)^(n+1)) / n^(4/3) and Σ[n=1 to ∞] ((-1)^n) / n! are both absolutely convergent. The first is an alternating series with a p > 1, and the second's terms decrease extremely quickly due to factorial growth in the denominator.

Step-by-step explanation:

The question asks which of the given series is absolutely convergent. A series is said to be absolutely convergent if the series of the absolute values of its terms is convergent. We will consider each series individually.

1. Σ[n=1 to ∞] ((-1)^(n+1)) / n^(4/3): This is an alternating series with decreasing absolute terms that converge to zero, and the function 1/n^(4/3) is eventually decreasing for all n. According to the Alternating Series Test, this series is convergent. To test absolute convergence, take the absolute value of the terms, resulting in Σ[n=1 to ∞] 1 / n^(4/3), which is a p-series with p = 4/3 > 1. Because p > 1, this series is absolutely convergent.
2. Σ[n=1 to ∞] ((-1)^n) / n!: This series is the expansion of the exponential function e^(-1) which converges. The factorial in the denominator grows much faster than the numerator, making the terms of the series approach zero very quickly. This series is not only convergent, but since factorial growth ensures that the terms decrease very rapidly, the sum of the absolute values also converges. Hence, this series is absolutely convergent.
3. The third series is not provided and therefore cannot be assessed for absolute convergence.

The series that is absolutely convergent are I and II, as both satisfy the conditions required for a series to be considered absolutely convergent.