Determining Convergence or Divergence
In Exercises 1–14, determine whether the alternating series converges or diverges. Some of the series do not satisfy the conditions of the Alternating Series Test.
∑ (from n = 1 to ∞) [(-1)ⁿ⁺¹ (1 / n^(3/2))]

Verified step by step guidance

1. Identify the general term of the series: \(a_n = \frac{1}{n^{3/2}}\) and the alternating factor is \((-1)^{n+1}\), which means the series is alternating in sign.
2. Check the conditions of the Alternating Series Test (Leibniz Test): (1) The terms \(a_n\) must be positive, (2) \(a_n\) must be decreasing, and (3) \(\lim_{n \to \infty} a_n = 0\).
3. Verify that \(a_n = \frac{1}{n^{3/2}}\) is positive for all \(n\) and that it decreases as \(n\) increases because the denominator grows larger.
4. Calculate the limit \(\lim_{n \to \infty} \frac{1}{n^{3/2}}\) to confirm it approaches zero, satisfying the third condition of the Alternating Series Test.
5. Since all conditions are met, conclude that the alternating series converges by the Alternating Series Test.

Key Concepts

Alternating Series Test

The Alternating Series Test determines convergence of series whose terms alternate in sign. It requires that the absolute value of terms decreases monotonically to zero. If these conditions hold, the series converges, even if it does not converge absolutely.

Absolute vs Conditional Convergence

A series converges absolutely if the series of absolute values converges. If the original alternating series converges but not absolutely, it is conditionally convergent. Understanding this distinction helps classify the nature of convergence.

p-Series and Convergence

A p-series has the form ∑ 1/n^p and converges if p > 1. In this problem, the terms involve 1/n^(3/2), where p = 3/2 > 1, indicating the absolute value series converges, which implies absolute convergence of the alternating series.