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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 10.3.87d
Chapter 10, Problem 10.3.87d

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


d. If ∑ pᵏ diverges, then ∑ (p + 0.001)ᵏ diverges, for a fixed real number p.

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1
Recall the behavior of geometric series: For a fixed real number \( r \), the series \( \sum r^k \) converges if and only if \( |r| < 1 \), and diverges otherwise.
Given that \( \sum p^k \) diverges, this means that \( |p| \geq 1 \). This is the key starting point for analyzing the series \( \sum (p + 0.001)^k \).
Consider the value of \( |p + 0.001| \). Since \( p \) is fixed, adding 0.001 shifts the base slightly. We need to determine if this new base still satisfies \( |p + 0.001| \geq 1 \) or not.
If \( |p + 0.001| \geq 1 \), then \( \sum (p + 0.001)^k \) also diverges by the geometric series test. However, if \( |p + 0.001| < 1 \), then \( \sum (p + 0.001)^k \) converges, providing a counterexample to the statement.
Therefore, to conclude whether the statement is true or false, analyze specific values of \( p \) where \( |p| = 1 \) and see how the small increment affects convergence or divergence.

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Key Concepts

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

Geometric Series and Convergence Criteria

A geometric series ∑ r^k converges if and only if the absolute value of the common ratio |r| is less than 1. If |r| ≥ 1, the series diverges. This criterion is fundamental to analyzing series of the form ∑ p^k and ∑ (p + 0.001)^k.
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Geometric Series

Effect of Changing the Common Ratio on Series Convergence

Altering the common ratio by a small amount (e.g., adding 0.001) can change whether the series converges or diverges. Even a slight increase can push the ratio beyond the convergence boundary, affecting the series' behavior significantly.
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Convergence of an Infinite Series

Counterexamples in Series Convergence

To determine the truth of a statement about series convergence, constructing counterexamples is essential. For instance, if ∑ p^k diverges but ∑ (p + 0.001)^k converges, this disproves the statement. Counterexamples help clarify the limits of general claims.
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Convergence of an Infinite Series
Related Practice
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.


43. ∑ (k = 1 to ∞) 1 / 3ᵏ

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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

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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. 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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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.

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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.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.

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