45–63. Absolute and conditional convergence Determine whether the following series converge absolutely, converge conditionally, or diverge.
∑ (k = 1 to ∞) (−1)ᵏ⁺¹ / (2√k − 1)

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1. Identify the given series: \( \sum_{k=1}^{\infty} \frac{(-1)^{k+1}}{2\sqrt{k} - 1} \). This is an alternating series because of the factor \( (-1)^{k+1} \), which alternates the sign of each term.
2. Check for absolute convergence by considering the series of absolute values: \( \sum_{k=1}^{\infty} \left| \frac{(-1)^{k+1}}{2\sqrt{k} - 1} \right| = \sum_{k=1}^{\infty} \frac{1}{2\sqrt{k} - 1} \). Determine if this series converges.
3. To analyze the absolute value series, compare \( \frac{1}{2\sqrt{k} - 1} \) to a simpler series. Since for large \( k \), \( 2\sqrt{k} - 1 \approx 2\sqrt{k} \), compare it to \( \frac{1}{\sqrt{k}} \), which is a p-series with \( p = \frac{1}{2} < 1 \) and diverges.
4. Since the absolute value series diverges, the original series does not converge absolutely. Next, apply the Alternating Series Test to the original series by checking if the terms \( b_k = \frac{1}{2\sqrt{k} - 1} \) decrease monotonically to zero.
5. Verify that \( b_k \) is positive, decreasing, and \( \lim_{k \to \infty} b_k = 0 \). If these conditions hold, the series converges conditionally by the Alternating Series Test.

Key Concepts

Absolute Convergence

A series ∑a_k converges absolutely if the series of absolute values ∑|a_k| converges. Absolute convergence guarantees convergence regardless of the sign of terms, and it implies the original series converges. Testing absolute convergence often involves comparison or root/ratio tests.

Conditional Convergence

A series converges conditionally if it converges, but does not converge absolutely. This typically occurs in alternating series where the terms decrease in magnitude to zero, but the absolute series diverges. The Alternating Series Test is commonly used to verify conditional convergence.

Alternating Series Test

The Alternating Series Test states that an alternating series ∑(−1)^{k} b_k converges if the sequence {b_k} is positive, decreasing, and approaches zero. This test helps determine conditional convergence when absolute convergence fails, especially for series with terms that alternate in sign.