Ch. 10 - Sequences and Infinite Series

Briggs - Calculus: Early Transcendentals 3rd Edition

Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook

Briggs 3rd Edition

Ch. 10 - Sequences and Infinite Series

Problem 10.2.97b

Chapter 10, Problem 10.2.97b

{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?

Verified step by step guidance

Understand the definition of the Fibonacci sequence given by the recurrence relation: \(f_{n+1} = f_n + f_{n-1}\) with initial conditions \(f_0 = 1\) and \(f_1 = 1\).

Recall that a sequence is bounded if there exists some real number \(M\) such that for all \(n\), \(|f_n| \leq M\).

Analyze the behavior of the Fibonacci sequence by considering the growth of its terms. Since each term is the sum of the two previous terms, the sequence tends to increase as \(n\) increases.

Recognize that the Fibonacci sequence grows approximately exponentially, as it can be expressed using Binet's formula involving powers of the golden ratio \(\phi = \frac{1 + \sqrt{5}}{2}\), which is greater than 1.

Conclude that because the terms grow without bound (they increase indefinitely), the Fibonacci sequence is not bounded.

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Key Concepts

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

Recurrence Relations

A recurrence relation defines each term of a sequence using previous terms. In the Fibonacci sequence, each term is the sum of the two preceding terms, starting with initial values. Understanding this helps analyze how the sequence evolves over time.

Boundedness of Sequences

A sequence is bounded if all its terms lie within some fixed interval. To determine if a sequence is bounded, one must check whether its terms stay below or above certain limits as the sequence progresses indefinitely.

Growth Behavior of the Fibonacci Sequence

The Fibonacci sequence grows exponentially because each term is roughly 1.618 times the previous term (the golden ratio). This means the terms increase without bound, indicating the sequence is unbounded.

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

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Textbook Question

27–34. Working with sequences Several terms of a sequence {aₙ}ₙ₌₁∞ are given.

b. Find a recurrence relation that generates the sequence (supply the initial value of the index and the first term of the sequence).

{1, 3, 9, 27, 81, ......}

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Textbook Question

{Use of Tech} Periodic dosing

Many people take aspirin on a regular basis as a preventive measure for heart disease. Suppose a person takes 80 mg of aspirin every 24 hours. Assume aspirin has a half-life of 24 hours; that is, every 24 hours, half of the drug in the blood is eliminated.

b.Use a calculator to estimate this limit. In the long run, how much drug is in the person’s blood?

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Textbook Question

{Use of Tech} Drug Dosing

A patient takes 75 mg of a medication every 12 hours; 60% of the medication in the blood is eliminated every 12 hours.

a.Let dₙ equal the amount of medication (in mg) in the bloodstream after n doses, where d₁ = 75.

Find a recurrence relation for dₙ.

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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.