Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.
d. When applying the Limit Comparison Test, an appropriate comparison series for ∑ (k = 1 to ∞) (k² + 2k + 1) / (k⁵ + 5k + 7) is ∑ (k = 1 to ∞) 1 / k³.

검증된 단계별 안내

Identify the general term of the given series: \(a_k = \frac{k^2 + 2k + 1}{k^5 + 5k + 7}\).

Determine the dominant terms in the numerator and denominator for large \(k\): numerator behaves like \(k^2\), denominator behaves like \(k^5\).

Simplify the behavior of \(a_k\) for large \(k\) by approximating it as \(\frac{k^2}{k^5} = \frac{1}{k^3}\).

Recall that the Limit Comparison Test involves comparing \(a_k\) with a known series \(b_k\) by evaluating \(\lim_{k \to \infty} \frac{a_k}{b_k}\).

Since \(a_k\) behaves like \(\frac{1}{k^3}\), choosing \(b_k = \frac{1}{k^3}\) is appropriate because the limit of \(\frac{a_k}{b_k}\) as \(k \to \infty\) will be a finite, positive number, satisfying the conditions of the Limit Comparison Test.

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주요 개념

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Limit Comparison Test

The Limit Comparison Test is used to determine the convergence or divergence of a series by comparing it to a second series with known behavior. It involves taking the limit of the ratio of the terms of the two series. If the limit is a positive finite number, both series either converge or diverge together.

Asymptotic Behavior of Series Terms

Analyzing the dominant terms in the numerator and denominator of a series term helps simplify the expression for large values of k. This simplification reveals the term's growth rate, which is crucial for choosing an appropriate comparison series in convergence tests.

p-Series and Their Convergence

A p-series has the form ∑ 1/k^p and converges if and only if p > 1. Recognizing whether a series behaves like a p-series for large k helps determine its convergence. In this problem, comparing to 1/k^3, a convergent p-series, is key to applying the Limit Comparison Test.

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.d. When applying the Limit Comparison Test, an appropriate comparison series for ∑ (k = 1 to ∞) (k² + 2k + 1) / (k⁵ + 5k + 7) is ∑ (k = 1 to ∞) 1 / k³.