Textbook Question
9–36. Comparison tests Use the Comparison Test or the Limit Comparison Test to determine whether the following series converge.
∑ (k = 1 to ∞) ((3k³ + 4)(7k² + 1)) / ((2k³ + 1)(4k³ − 1))
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14. Sequences & Series
Convergence Tests
3:13 minutesProblem 10.5.11
Textbook Question
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)
Verified step by step guidance1
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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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
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.
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Direct Comparison Test
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.
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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.
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