77–87. Absolute or conditional convergence
Determine whether the following series converge absolutely, converge conditionally, or diverge.
∑ (from k = 1 to ∞)(−1)ᵏ⁺¹ / k³⁄⁷

Verified step by step guidance

1. Identify the given series: \( \sum_{k=1}^{\infty} \frac{(-1)^{k+1}}{k^{3/7}} \). This is an alternating series because of the factor \( (-1)^{k+1} \), which alternates the sign of each term.
2. Check for absolute convergence by considering the series of absolute values: \( \sum_{k=1}^{\infty} \left| \frac{(-1)^{k+1}}{k^{3/7}} \right| = \sum_{k=1}^{\infty} \frac{1}{k^{3/7}} \). This is a p-series with \( p = \frac{3}{7} \).
3. Recall that a p-series \( \sum \frac{1}{k^p} \) converges if and only if \( p > 1 \). Since \( \frac{3}{7} < 1 \), the series of absolute values diverges, so the original series does not converge absolutely.
4. Next, apply the Alternating Series Test to the original series. Check if the terms \( b_k = \frac{1}{k^{3/7}} \) are positive, decreasing, and approach zero as \( k \to \infty \).
5. Since \( b_k \) is positive, decreasing, and \( \lim_{k \to \infty} b_k = 0 \), the Alternating Series Test confirms that the original series converges conditionally.

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 p-series tests.

Conditional Convergence

A series converges conditionally if it converges, but does not converge absolutely. This means ∑a_k converges, but ∑|a_k| diverges. Alternating series with terms decreasing to zero often exhibit conditional convergence, which requires specific 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 3/7 is less than 1, so the absolute series diverges. The Alternating Series Test can then be used to check if the original alternating series converges conditionally.