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Ch. 10 - Sequences and Infinite Series
Briggs - Calculus: Early Transcendentals 3rd Edition
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Tutti i libri di testoBriggs 3rd EditionCh. 10 - Sequences and Infinite SeriesProblema 10.5.37d
Capitolo 10, Problema 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.

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1
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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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.
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07:45
Limit Comparison Test

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.
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Geometric Series

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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04:30
P-Series and Harmonic Series
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Domanda del libro di testo

41–44. {Use of Tech} Remainders and estimates Consider the following convergent series.


d. Find an interval in which the value of the series must lie if you approximate it using ten terms of the series.


43. ∑ (k = 1 to ∞) 1 / 3ᵏ

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Domanda del libro di testo

{Use of Tech} A savings plan

James begins a savings plan in which he deposits \$100 at the beginning of each month into an account that earns 9% interest annually, or equivalently, 0.75% per month.

To be clear, on the first day of each month, the bank adds 0.75% of the current balance as interest, and then James deposits \$100.


Let Bₙ be the balance in the account after the nᵗʰ payment, where B₀ = \$0.


c.How many months are needed to reach a balance of \$5000?

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Domanda del libro di testo

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


c. If lim (as k → ∞) ᵏ√|aₖ| = 1/4, then ∑ 10aₖ converges absolutely.

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Domanda del libro di testo

72–75. {Use of Tech} Practical sequences

Consider the following situations that generate a sequence


d.Using a calculator or a graphing utility, estimate the limit of the sequence or state that it does not exist.


Radioactive decay

A material transmutes 50% of its mass to another element every 10 years due to radioactive decay. Let Mₙ be the mass of the radioactive material at the end of the nᵗʰ decade, where the initial mass of the material is M₀ = 20g.

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Domanda del libro di testo

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

d. If ∑ aₖ diverges, then ∑ |aₖ| diverges.

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Domanda del libro di testo

41–44. {Use of Tech} Remainders and estimates Consider the following convergent series.


c. Find lower and upper bounds (Lₙ and Uₙ, respectively) on the exact value of the series.


43. ∑ (k = 1 to ∞) 1 / 3ᵏ

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