Ch. 10 - Sequences and Infinite Series

Briggs - Calculus: Early Transcendentals 3rd Edition

Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Non è quello che usi tu?Cambia libro di testo

Briggs 3rd Edition

Ch. 10 - Sequences and Infinite Series

Problema 10.2.97b

Capitolo 10, Problema 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?

Guida verificata passo dopo passo

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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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.

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Intro To Related Rates

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.

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Introduction to Sequences

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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Introduction to Sequences

Pratica correlata

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{Use of Tech} A savings plan

James begins a savings plan in which he deposits \$100 at the beginning of each month into an account that earns 9% interest annually, or equivalently, 0.75% per month.

To be clear, on the first day of each month, the bank adds 0.75% of the current balance as interest, and then James deposits \$100.

Let Bₙ be the balance in the account after the nᵗʰ payment, where B₀ = \$0.

b.Find a recurrence relation that generates the sequence {Bₙ}.

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Explain why or why not

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

b.If limₙ→∞aₙ = 0 and limₙ→∞bₙ = ∞, then limₙ→∞aₙbₙ = 0.

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41–44. {Use of Tech} Remainders and estimates Consider the following convergent series.

b. Find how many terms are needed to ensure that the remainder is less than 10⁻³.

41. ∑ (k = 1 to ∞) 1 / k⁶

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{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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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, 2, 4, 8, 16, ......}

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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 = 30,r = 0.25

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