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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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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

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

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.

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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Textbook Question

71. Evaluating an infinite series two ways

Evaluate the series

∑ (k = 1 to ∞) (4 / 3ᵏ – 4 / 3ᵏ⁺¹) two ways.


b. Use a geometric series argument with Theorem 10.8.

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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. The sum ∑ (k = 3 to ∞) 1 / √(k − 2) is a p-series.

Textbook Question

{Use of Tech} Fibonacci sequence

The famous Fibonacci sequence was proposed by Leonardo Pisano, also known as Fibonacci, in about A.D. 1200 as a model for the growth of rabbit populations.


It is given by the recurrence relation: fₙ₊₁ = fₙ + fₙ₋₁,for n = 1, 2, 3, … where f₀ = 1 and f₁ = 1. Each term of the sequence is the sum of its two predecessors. 


b.Is the sequence bounded?

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