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Ch. 10 - Sequences and Infinite Series
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 10 - Sequences and Infinite SeriesProblem 10.4.47d
Chapter 10, Problem 10.4.47d

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.

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1
Recall that the series \( \sum_{k=1}^\infty \frac{1}{k^2} \) is a convergent series with positive terms, known as the p-series with \( p = 2 > 1 \).
The partial sum \( S_n = \sum_{k=1}^n \frac{1}{k^2} \) represents the sum of the first \( n \) terms of the series.
Since all terms \( \frac{1}{k^2} > 0 \), the sequence of partial sums \( S_n \) is strictly increasing, meaning \( S_1 < S_2 < S_3 < \cdots \).
Because the series converges to a finite limit \( S = \sum_{k=1}^\infty \frac{1}{k^2} \), and the partial sums increase towards this limit, each partial sum \( S_n \) must be less than or equal to \( S \).
Therefore, every partial sum \( S_n \) underestimates the exact value of the infinite series \( \sum_{k=1}^\infty \frac{1}{k^2} \), making the statement true.

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Key Concepts

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

Partial Sums of Infinite Series

A partial sum Sₙ is the sum of the first n terms of an infinite series. It approximates the total sum, and understanding how these sums behave helps determine if they overestimate or underestimate the series' exact value.
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Intro to Series: Partial Sums

Convergence and Monotonicity of Series

A series converges if its partial sums approach a finite limit. For series with positive, decreasing terms like 1/k², partial sums are increasing and approach the limit from below, which affects whether they underestimate or overestimate the total sum.
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Convergence of an Infinite Series

Comparison and Counterexamples in Series Analysis

To verify statements about series, one uses comparison tests or constructs counterexamples. For the series ∑ 1/k², known results and inequalities help confirm if partial sums underestimate the total sum or not.
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Geometric Series
Related Practice
Textbook Question

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.


Drug elimination

Jack took a 200-mg dose of a pain killer at midnight. Every hour, 5% of the drug is washed out of his bloodstream. Let dₙ be the amount of drug in Jack’s blood n hours after the drug was taken, where d₀ = 200mg.

Textbook Question

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.


41. ∑ (k = 1 to ∞) 1 / k⁶

Textbook Question

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.

Textbook Question

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.

Textbook Question

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.

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.