9–36. Comparison tests Use the Comparison Test or the Limit Comparison Test to determine whether the following series converge.

∑ (k = 1 to ∞) (k² − 1) / (k³ + 4)

검증된 단계별 안내

Identify the given series: \( \sum_{k=1}^{\infty} \frac{k^{2} - 1}{k^{3} + 4} \). We want to determine if this series converges or diverges.

Analyze the behavior of the general term \( a_k = \frac{k^{2} - 1}{k^{3} + 4} \) for large \( k \). For large \( k \), the dominant terms in numerator and denominator are \( k^{2} \) and \( k^{3} \), respectively, so \( a_k \) behaves like \( \frac{k^{2}}{k^{3}} = \frac{1}{k} \).

Choose a comparison series that is simpler but has similar behavior for large \( k \). Since \( a_k \) behaves like \( \frac{1}{k} \), consider the harmonic series \( \sum_{k=1}^{\infty} \frac{1}{k} \), which is known to diverge.

Apply the Limit Comparison Test by computing the limit \( L = \lim_{k \to \infty} \frac{a_k}{b_k} = \lim_{k \to \infty} \frac{\frac{k^{2} - 1}{k^{3} + 4}}{\frac{1}{k}} = \lim_{k \to \infty} \frac{(k^{2} - 1) \cdot k}{k^{3} + 4} \). Simplify this expression to find \( L \).

Interpret the result of the limit \( L \). If \( L \) is a finite positive number, then both series either converge or diverge together. Since the harmonic series diverges, this will tell us about the convergence of the original series.

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주요 개념

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Comparison Test

The Comparison Test determines the convergence of a series by comparing it to a second series with known behavior. If the terms of the given series are smaller than those of a convergent series, it also converges. Conversely, if the terms are larger than those of a divergent series, it diverges.

Limit Comparison Test

The Limit Comparison Test compares two series by taking the limit of the ratio of their terms. If this limit is a positive finite number, both series either converge or diverge together. This test is useful when direct comparison is difficult but the terms behave similarly for large indices.

Behavior of Rational Functions in Series

When analyzing series with rational function terms, focus on the highest degree terms in numerator and denominator to determine the dominant behavior as k approaches infinity. This helps identify a simpler comparison series, often a p-series or geometric series, to apply convergence tests effectively.

9–36. Comparison tests Use the Comparison Test or the Limit Comparison Test to determine whether the following series converge.∑ (k = 1 to ∞) (k² − 1) / (k³ + 4)