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 / √(k³ᐟ² + k)

검증된 단계별 안내

Identify the given series: \( \sum_{k=1}^{\infty} \frac{(-1)^k}{\sqrt{k^{3/2} + k}} \). Notice it is an alternating series because of the factor \( (-1)^k \).

Check for absolute convergence by considering the absolute value of the terms: \( \left| \frac{(-1)^k}{\sqrt{k^{3/2} + k}} \right| = \frac{1}{\sqrt{k^{3/2} + k}} \).

Simplify the denominator inside the square root to understand the behavior for large \( k \): \( \sqrt{k^{3/2} + k} = \sqrt{k^{3/2}(1 + k^{1/2 - 1})} = \sqrt{k^{3/2}} \sqrt{1 + k^{-1/2}} \). Since \( \sqrt{k^{3/2}} = k^{3/4} \), the term behaves like \( \frac{1}{k^{3/4}} \) for large \( k \).

Determine if the series of absolute values \( \sum \frac{1}{k^{3/4}} \) converges. Recall that the p-series \( \sum \frac{1}{k^p} \) converges if and only if \( p > 1 \). Here, \( p = \frac{3}{4} < 1 \), so the series does not converge absolutely.

Since the series is alternating, apply the Alternating Series Test: check if the terms \( b_k = \frac{1}{\sqrt{k^{3/2} + k}} \) decrease monotonically to zero. If they do, the series converges conditionally.

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주요 개념

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Absolute and Conditional Convergence

Absolute convergence occurs when the series of absolute values converges, ensuring the original series converges regardless of term signs. Conditional convergence happens when the series converges, but the series of absolute values diverges. Distinguishing between these helps classify the behavior of alternating series.

Alternating Series Test

The Alternating Series Test determines convergence of series whose terms alternate in sign. If the absolute value of terms decreases monotonically to zero, the series converges. This test is useful for series with factors like (−1)^k, as in the given problem.

Comparison and p-Series Tests

Comparison tests involve comparing a given series to a known benchmark series to determine convergence. The p-series test states that ∑ 1/k^p converges if p > 1 and diverges otherwise. Recognizing the dominant term in the denominator helps apply these tests effectively.

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 / √(k³ᐟ² + k)