45–63. Absolute and conditional convergence Determine whether the following series converge absolutely, converge conditionally, or diverge.
∑ (k = 1 to ∞) (−1)ᵏ / k^(2/3)

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Identify the given series: \( \sum_{k=1}^{\infty} \frac{(-1)^k}{k^{2/3}} \). This is an alternating series because of the factor \( (-1)^k \), which alternates the sign of each term.

Check for absolute convergence by considering the series of absolute values: \( \sum_{k=1}^{\infty} \left| \frac{(-1)^k}{k^{2/3}} \right| = \sum_{k=1}^{\infty} \frac{1}{k^{2/3}} \).

Determine if the series \( \sum_{k=1}^{\infty} \frac{1}{k^{2/3}} \) converges. This is a p-series with \( p = \frac{2}{3} \). Recall that a p-series \( \sum \frac{1}{k^p} \) converges if and only if \( p > 1 \). Since \( \frac{2}{3} < 1 \), this series diverges.

Since the series does not converge absolutely, check for conditional convergence by applying the Alternating Series Test. The test requires that the terms \( b_k = \frac{1}{k^{2/3}} \) are positive, decreasing, and approach zero as \( k \to \infty \).

Verify that \( b_k = \frac{1}{k^{2/3}} \) is positive, decreasing, and \( \lim_{k \to \infty} b_k = 0 \). Since these conditions hold, the original alternating series converges conditionally by the Alternating Series Test.

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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 implies the original series converges. Testing absolute convergence often involves comparison or p-series tests.

Conditional Convergence

A series converges conditionally if it converges, but does not converge absolutely. This means the series ∑a_k converges, but ∑|a_k| diverges. Alternating series with terms decreasing to zero often exhibit conditional convergence, which requires careful analysis using tests like the Alternating Series Test.

p-Series and Convergence Tests

A p-series ∑1/k^p converges if p > 1 and diverges otherwise. For the given series, the exponent 2/3 < 1, so the corresponding positive term series diverges. This fact helps determine absolute convergence and guides the use of the Alternating Series Test for conditional convergence.

45–63. Absolute and conditional convergence Determine whether the following series converge absolutely, converge conditionally, or diverge.∑ (k = 1 to ∞) (−1)ᵏ / k^(2/3)