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

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Identify the given series: \( \sum_{k=1}^{\infty} (-1)^k \frac{\tan^{-1}(k)}{k^3} \). This is an alternating series because of the factor \( (-1)^k \).

Check the absolute convergence by considering the series of absolute values: \( \sum_{k=1}^{\infty} \left| \frac{\tan^{-1}(k)}{k^3} \right| = \sum_{k=1}^{\infty} \frac{\tan^{-1}(k)}{k^3} \). Since \( \tan^{-1}(k) \) is positive for all positive \( k \), the absolute value removes the alternating sign.

Analyze the behavior of \( \tan^{-1}(k) \) as \( k \to \infty \). Recall that \( \tan^{-1}(k) \) approaches \( \frac{\pi}{2} \), a finite constant, so for large \( k \), \( \frac{\tan^{-1}(k)}{k^3} \) behaves like \( \frac{\pi/2}{k^3} \).

Compare the absolute value series to the p-series \( \sum \frac{1}{k^3} \), which converges because \( p = 3 > 1 \). By the Comparison Test, the absolute value series converges.

Since the series converges absolutely, conclude that the original alternating series converges absolutely as well.

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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 often simplifies analysis by allowing the use of comparison tests on positive terms.

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 and Behavior of arctan(k)/k³

The Alternating Series Test states that if the absolute value of terms decreases monotonically to zero, the alternating series converges. Here, tan⁻¹(k)/k³ decreases to zero as k grows large, so this test helps determine conditional convergence by analyzing term behavior and sign alternation.

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