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

Verified step by step guidance

1. Identify the given series: \( \sum_{k=1}^{\infty} (-1)^k \cdot \frac{k}{2k + 1} \). This is an alternating series because of the factor \( (-1)^k \).
2. Check for absolute convergence by considering the absolute value of the terms: \( \sum_{k=1}^{\infty} \left| (-1)^k \cdot \frac{k}{2k + 1} \right| = \sum_{k=1}^{\infty} \frac{k}{2k + 1} \).
3. Analyze the behavior of the absolute value terms \( \frac{k}{2k + 1} \) as \( k \to \infty \). Determine if the series \( \sum \frac{k}{2k + 1} \) converges or diverges by comparing it to a simpler series.
4. If the absolute value series diverges, test the original alternating series for conditional convergence using the Alternating Series Test. Verify if the sequence \( b_k = \frac{k}{2k + 1} \) is decreasing and if \( \lim_{k \to \infty} b_k = 0 \).
5. Based on the results from the absolute convergence test and the Alternating Series Test, conclude whether the series converges absolutely, converges conditionally, or diverges.

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 ratio tests.

Conditional Convergence

A series converges conditionally if it converges, but does not converge absolutely. This means ∑a_k converges, but ∑|a_k| diverges. Conditional convergence often occurs in alternating series where the terms decrease in magnitude and approach zero.

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.