77–87. Absolute or conditional convergence
Determine whether the following series converge absolutely, converge conditionally, or diverge.
∑ (from k = 1 to ∞)(−1)ᵏ⁺¹(k² + 4) / (2k² + 1)

Verified step by step guidance

1. Identify the given series: \( \sum_{k=1}^{\infty} (-1)^{k+1} \frac{k^2 + 4}{2k^2 + 1} \). This is an alternating series because of the factor \( (-1)^{k+1} \).
2. Check for absolute convergence by considering the series of absolute values: \( \sum_{k=1}^{\infty} \left| (-1)^{k+1} \frac{k^2 + 4}{2k^2 + 1} \right| = \sum_{k=1}^{\infty} \frac{k^2 + 4}{2k^2 + 1} \).
3. Analyze the behavior of the terms \( \frac{k^2 + 4}{2k^2 + 1} \) as \( k \to \infty \). Simplify the expression by dividing numerator and denominator by \( k^2 \) to find the limit of the terms.
4. Determine if the series of absolute values converges or diverges by comparing it to a known benchmark series, such as a \( p \)-series or using the Limit Comparison Test.
5. If the series of absolute values diverges, apply the Alternating Series Test to the original series by checking if the terms \( \frac{k^2 + 4}{2k^2 + 1} \) decrease monotonically to zero. Based on this, conclude whether the original series 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 implies convergence regardless of the sign of terms, and it guarantees the sum is well-defined and stable under rearrangement.

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 but the absolute series diverges.

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.