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 10.1.75d

Chapter 10, Problem 10.1.75d

72–75. {Use of Tech} Practical sequences
Consider the following situations that generate a sequence

d.Using a calculator or a graphing utility, estimate the limit of the sequence or state that it does not exist.

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.

Verified step by step guidance

Identify the type of sequence described. Since 5% of the drug is eliminated every hour, 95% remains each hour. This means the amount of drug decreases by a constant ratio, indicating a geometric sequence.

Write the general formula for the amount of drug after n hours. Since the initial amount is \(d_0 = 200\) mg and the drug decreases by 5% each hour, the formula is: \(d_n = 200 \times (0.95)^n\)

Understand the behavior of the sequence as \(n\) becomes very large. Since \(0 < 0.95 < 1\), the term \((0.95)^n\) gets smaller and smaller as \(n\) increases.

Use a calculator or graphing utility to evaluate \(d_n\) for large values of \(n\) to observe the trend. This will help estimate the limit of the sequence.

Conclude the limit based on the observed behavior: if \(d_n\) approaches a specific value as \(n\) approaches infinity, that value is the limit; otherwise, state that the limit does not exist.

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Key Concepts

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

Geometric Sequences

A geometric sequence is a sequence where each term is found by multiplying the previous term by a constant ratio. In this problem, the amount of drug decreases by a fixed percentage each hour, forming a geometric sequence with a common ratio less than 1.

Limit of a Sequence

The limit of a sequence describes the value that the terms approach as the index goes to infinity. For a geometric sequence with a ratio between -1 and 1, the limit exists and can be used to understand the long-term behavior of the drug concentration.

Use of Technology in Calculus

Calculators or graphing utilities can approximate sequence values and limits when analytical solutions are complex. They help visualize the sequence behavior over time and estimate limits, especially useful for real-world applications like drug elimination.

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Related Practice

Textbook Question

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

d. Find an interval in which the value of the series must lie if you approximate it using ten terms of the series.

43. ∑ (k = 1 to ∞) 1 / 3ᵏ

137

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Textbook Question

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

c.Make a conjecture for the value of the series.

∑⁽∞⁾ₖ₌₁2⁄[(2k − 1)(2k + 1)]

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Textbook Question

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

c. If lim (as k → ∞) ᵏ√|aₖ| = 1/4, then ∑ 10aₖ converges absolutely.

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Textbook Question

72–75. {Use of Tech} Practical sequences

Consider the following situations that generate a sequence

d.Using a calculator or a graphing utility, estimate the limit of the sequence or state that it does not exist.

Radioactive decay

A material transmutes 50% of its mass to another element every 10 years due to radioactive decay. Let Mₙ be the mass of the radioactive material at the end of the nᵗʰ decade, where the initial mass of the material is M₀ = 20g.

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Textbook Question

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

d. If ∑ aₖ diverges, then ∑ |aₖ| diverges.

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Textbook Question

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

c. Find lower and upper bounds (Lₙ and Uₙ, respectively) on the exact value of the series.

43. ∑ (k = 1 to ∞) 1 / 3ᵏ

147

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