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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 10.1.71b
Chapter 10, Problem 10.1.71b

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


b.If a sequence of positive numbers converges, then the sequence is decreasing.

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1
Recall the definition of a sequence converging: A sequence \( \{a_n\} \) converges to a limit \( L \) if for every \( \epsilon > 0 \), there exists an \( N \) such that for all \( n > N \), \( |a_n - L| < \epsilon \).
Understand what it means for a sequence to be decreasing: A sequence \( \{a_n\} \) is decreasing if \( a_{n+1} \leq a_n \) for all \( n \).
Analyze the statement: "If a sequence of positive numbers converges, then the sequence is decreasing." This implies that every convergent sequence of positive terms must be decreasing.
Consider a counterexample to test the statement: For instance, the sequence \( a_n = \frac{1}{n} \) is positive and converges to 0, and it is decreasing. However, the sequence \( a_n = \frac{1}{n} \) is decreasing, but what about a sequence like \( a_n = \frac{1}{n} \) for odd \( n \) and \( a_n = \frac{1}{n} + \frac{1}{n^2} \) for even \( n \)? This sequence is positive and converges to 0 but is not strictly decreasing.
Conclude that convergence of a positive sequence does not guarantee it is decreasing, so the statement is false. A convergent sequence can oscillate or increase at some points as long as it approaches the limit.

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Key Concepts

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

Convergence of a Sequence

A sequence converges if its terms approach a specific finite limit as the index goes to infinity. Convergence does not impose restrictions on the sequence's monotonicity; the terms can oscillate or increase before settling near the limit.
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Monotonic Sequences

A sequence is decreasing if each term is less than or equal to the previous term. Monotonicity is a property describing the sequence's order, independent of whether it converges or not.
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Counterexamples in Sequence Analysis

To disprove a statement about sequences, providing a counterexample—a sequence that meets the conditions but violates the conclusion—is effective. For instance, a convergent sequence of positive numbers that is not decreasing shows the statement is false.
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Related Practice
Textbook Question

18–20. Evaluating geometric series two ways Evaluate each geometric series two ways.


a. Find the nth partial sum Sₙ of the series and evaluate lim (as n → ∞) Sₙ.


∑ (k = 0 to ∞) (–2/7)ᵏ

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Textbook Question

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


b. (2n)! / (2n − 1)! = 2n

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Textbook Question

67–70. Formulas for sequences of partial sums Consider the following infinite series.


a.Find the first four partial sums S₁, S₂, S₃, S₄ of the series.


∑⁽∞⁾ₖ₌₁2⁄[(2k − 1)(2k + 1)]

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Textbook Question

57–60. Heights of bouncing balls A ball is thrown upward to a height of hₒ meters. After each bounce, the ball rebounds to a fraction r of its previous height. Let hₙ be the height after the nth bounce. Consider the following values of hₒ and r.


b. Find an explicit formula for the nth term of the sequence {hₙ}.


h₀ = 20,r = 0.5

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Textbook Question

87. Explain why or why not

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


b. If ∑ (k = 1 to ∞) aₖ diverges, then ∑ (k = 10 to ∞) aₖ diverges.

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Textbook Question

72–75. {Use of Tech} Practical sequences

Consider the following situations that generate a sequence


b.Find an explicit formula for the terms of the sequence.


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.

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