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Ch. 10 - Sequences and Infinite Series
Briggs - Calculus: Early Transcendentals 3rd Edition
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Tutti i libri di testoBriggs 3rd EditionCh. 10 - Sequences and Infinite SeriesProblema 10.2.93b
Capitolo 10, Problema 10.2.93b

Suppose the sequence {aₙ}⁽∞⁾ₙ₌₀ is defined by the recurrence relation
aₙ₊₁ = ⅓aₙ + 6;a₀ = 3.


b.Explain why {aₙ}⁽∞⁾ₙ₌₀ converges and find the limit.

Guida verificata passo dopo passo
1
Recognize that the sequence \( \{a_n\} \) is defined by a linear recurrence relation of the form \( a_{n+1} = r a_n + c \), where \( r = \frac{1}{3} \) and \( c = 6 \). Since \( |r| = \frac{1}{3} < 1 \), the sequence is a contraction and will converge to a fixed point.
To find the limit \( L \) of the sequence, assume it exists and satisfies the recurrence relation in the limit, so \( L = \frac{1}{3} L + 6 \). This is because as \( n \to \infty \), \( a_n \to L \) and \( a_{n+1} \to L \).
Solve the equation for \( L \): \( L = \frac{1}{3} L + 6 \). Rearranging gives \( L - \frac{1}{3} L = 6 \), which simplifies to \( \frac{2}{3} L = 6 \).
Multiply both sides by \( \frac{3}{2} \) to isolate \( L \), yielding \( L = 6 \times \frac{3}{2} \).
Interpret the result: since the sequence converges to \( L \), this value is the fixed point of the recurrence relation and represents the long-term behavior of \( \{a_n\} \).

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Recurrence Relations

A recurrence relation defines each term of a sequence based on previous terms. Understanding how to manipulate and solve these relations is essential to analyze the behavior of sequences, such as finding explicit formulas or limits.
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Intro To Related Rates

Convergence of Sequences

A sequence converges if its terms approach a specific finite value as the index goes to infinity. Determining convergence involves analyzing the long-term behavior of the sequence, often by examining the recurrence relation or using limit properties.
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Introduction to Sequences

Finding Limits of Linear Recurrence Sequences

For linear recurrence relations like aₙ₊₁ = r aₙ + c with |r| < 1, the sequence converges to the fixed point L = c / (1 - r). This limit is found by setting aₙ₊₁ = aₙ = L and solving the resulting equation.
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Finding Limits by Direct Substitution
Pratica correlata
Domanda del libro di testo

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


b.Find a formula for the nth partial sum Sₙ of the infinite series. Use this formula to find the next four partial sums S₅, S₆, S₇, S₈ of the infinite series.


∑⁽∞⁾ₖ₌₁2⁄[(2k − 1)(2k + 1)]

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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.


43. ∑ (k = 1 to ∞) 1 / 3ᵏ

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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, 3, 9, 27, 81, ......}

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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 = 20,r = 0.5

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


b. Evaluate the series using Theorem 10.7.


∑ (k = 0 to ∞) (–2/7)ᵏ

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