Ch. 10 - Sequences and Infinite Series

Briggs - Calculus: Early Transcendentals 3rd Edition

Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook

Briggs 3rd Edition

Ch. 10 - Sequences and Infinite Series

Problem 10.5.37d

Chapter 10, Problem 10.5.37d

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.

Verified step by step guidance

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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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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.

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P-Series and Harmonic Series

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Textbook Question

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41. ∑ (k = 1 to ∞) 1 / k⁶

Textbook Question

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Textbook Question

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.

d. The Ratio Test is always inconclusive when applied to ∑ aₖ, where aₖ is a nonzero rational function of k.

Textbook Question

Explain why or why not

Determine whether the following statements are true and give an explanation or counterexample.

d.If {aₙ} = {1, ½, ⅓, ¼, ⅕, …} and

{bₙ} = {1, 0, ½, 0, ⅓, 0, ¼, 0, …},

then limₙ→∞aₙ = limₙ→∞bₙ.

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Textbook Question

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.

d. Every partial sum Sₙ of the series ∑ (k = 1 to ∞) 1 / k² underestimates the exact value of ∑ (k = 1 to ∞) 1 / k².

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.

Verified video answer for a similar problem:

Key Concepts

Limit Comparison Test

Asymptotic Behavior of Series Terms

p-Series and Their Convergence

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.