45–63. Absolute and conditional convergence Determine whether the following series converge absolutely, converge conditionally, or diverge.
∑ (k = 2 to ∞) (−1)ᵏ · k · (k² + 1) / (k³ − 1)

Verified step by step guidance

1. Identify the given series: \( \sum_{k=2}^{\infty} (-1)^k \cdot k \cdot \frac{k^2 + 1}{k^3 - 1} \). Notice it is an alternating series because of the factor \( (-1)^k \).
2. To check for absolute convergence, consider the absolute value of the terms: \( \left| (-1)^k \cdot k \cdot \frac{k^2 + 1}{k^3 - 1} \right| = k \cdot \frac{k^2 + 1}{k^3 - 1} \). Simplify this expression to understand its behavior as \( k \to \infty \).
3. Analyze the limit of the absolute value terms as \( k \to \infty \). Since the highest powers dominate, approximate \( k \cdot \frac{k^2 + 1}{k^3 - 1} \approx k \cdot \frac{k^2}{k^3} = 1 \). This suggests the terms do not approach zero, which is crucial for convergence.
4. Since the terms of the absolute value do not tend to zero, the series does not converge absolutely. Next, check for conditional convergence by applying the Alternating Series Test (Leibniz Test). For this, verify if the sequence \( a_k = k \cdot \frac{k^2 + 1}{k^3 - 1} \) is decreasing and tends to zero.
5. Determine whether \( a_k \) is decreasing and if \( \lim_{k \to \infty} a_k = 0 \). If either condition fails, the series diverges; if both hold, the series converges conditionally.

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.

Tests for Convergence of Series

To determine convergence, tests like the Alternating Series Test, Comparison Test, or Limit Comparison Test are used. For the given series, analyzing the behavior of terms and applying these tests helps decide if the series converges absolutely, conditionally, or diverges.