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))

Verified step by step guidance

1. First, identify the general term of the series: \(a_k = \frac{(3k^{3} + 4)(7k^{2} + 1)}{(2k^{3} + 1)(4k^{3} - 1)}\).
2. Next, analyze the behavior of \(a_k\) for large \(k\) by focusing on the highest degree terms in numerator and denominator: numerator behaves like \$3k^{3} \times 7k^{2} = 21k^{5}\(, denominator behaves like \)2k^{3} \times 4k^{3} = 8k^{6}$.
3. Simplify the dominant term ratio for large \(k\): \(a_k \sim \frac{21k^{5}}{8k^{6}} = \frac{21}{8k}\), which suggests \(a_k\) behaves like \(\frac{C}{k}\) for some constant \(C\) as \(k \to \infty\).
4. Choose a comparison series \(b_k = \frac{1}{k}\), which is a \(p\)-series with \(p=1\), known to diverge.
5. Apply the Limit Comparison Test by computing \(\lim_{k \to \infty} \frac{a_k}{b_k} = \lim_{k \to \infty} a_k \times k\), and analyze the limit to determine if it is finite and positive, which will indicate whether \(\sum a_k\) converges or diverges.

Key Concepts

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. This test requires finding a suitable comparison series.

Limit Comparison Test

The Limit Comparison Test involves taking the limit of the ratio of the terms of two series. 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.

Asymptotic Behavior of Rational Functions

Analyzing the dominant terms in the numerator and denominator of rational expressions helps simplify the general term of a series. For large k, lower-degree terms become negligible, allowing approximation of the term’s behavior. This simplification is crucial for choosing an appropriate comparison series.