32–49. Choose your test Use the test of your choice to determine whether the following series converge absolutely, converge conditionally, or diverge.
∑ (from k = 1 to ∞) (−1)ᵏ / k⁰.⁹⁹

검증된 단계별 안내

Identify the given series: \( \sum_{k=1}^{\infty} \frac{(-1)^k}{k^{0.99}} \). This is an alternating series because of the factor \( (-1)^k \).

Check if the series converges absolutely by considering the absolute value of the terms: \( \sum_{k=1}^{\infty} \left| \frac{(-1)^k}{k^{0.99}} \right| = \sum_{k=1}^{\infty} \frac{1}{k^{0.99}} \).

Determine whether the series \( \sum_{k=1}^{\infty} \frac{1}{k^{0.99}} \) converges. Recall that the p-series \( \sum \frac{1}{k^p} \) converges if and only if \( p > 1 \). Since \( 0.99 < 1 \), this series diverges, so the original series does not converge absolutely.

Apply the Alternating Series Test (Leibniz Test) to the original series. Check two conditions: (1) the terms \( b_k = \frac{1}{k^{0.99}} \) decrease monotonically, and (2) \( \lim_{k \to \infty} b_k = 0 \). Both conditions hold here.

Conclude that since the series converges by the Alternating Series Test but does not converge absolutely, it converges conditionally.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Absolute Convergence

A series ∑a_k converges absolutely if the series of absolute values ∑|a_k| converges. Absolute convergence guarantees convergence regardless of the signs of the terms. 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 typically occurs in alternating series where the terms decrease in magnitude to zero, but the absolute series diverges. The Alternating Series Test is commonly used to verify this.

Alternating Series Test

This test determines convergence of series whose terms alternate in sign. If the absolute value of terms decreases monotonically to zero, the series converges. It does not guarantee absolute convergence, only conditional convergence.

32–49. Choose your test Use the test of your choice to determine whether the following series converge absolutely, converge conditionally, or diverge.∑ (from k = 1 to ∞) (−1)ᵏ / k⁰.⁹⁹