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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 10.4.47a
Chapter 10, Problem 10.4.47a

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


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

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1
Recall the definition of a p-series: a series of the form \(\sum_{k=1}^{\infty} \frac{1}{k^p}\), where \(p\) is a positive real number.
Examine the given series: \(\sum_{k=1}^{\infty} \frac{1}{3^k}\). Notice that the denominator is \$3^k\(, which is an exponential expression, not a power of \)k$.
Since the denominator is not of the form \(k^p\), the series does not fit the definition of a p-series.
Instead, the given series is a geometric series with common ratio \(r = \frac{1}{3}\), because each term is obtained by multiplying the previous term by \(\frac{1}{3}\).
Therefore, the statement that the series is a p-series is false; it is actually a geometric series.

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Key Concepts

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

Definition of a p-series

A p-series is an infinite series of the form ∑ 1/n^p, where n starts at 1 and p is a positive constant. It converges if and only if p > 1. Recognizing whether a series fits this form is essential to classify it as a p-series.
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P-Series and Harmonic Series

Geometric series

A geometric series is an infinite sum where each term is a constant ratio r times the previous term, expressed as ∑ ar^k. It converges if |r| < 1, and its sum can be calculated using the formula a/(1-r). Identifying geometric series helps distinguish them from p-series.
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Geometric Series

Series classification and counterexamples

Determining whether a series belongs to a specific type requires comparing its general term to the defining form. Providing counterexamples or explanations clarifies why a series does or does not fit a category, aiding in understanding convergence properties.
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Geometric Series
Related Practice
Textbook Question

{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.


a.Write the first five terms of the sequence {Bₙ}.

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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

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

a. Find the next two terms of the sequence.

{1, 2, 4, 8, 16, ......}

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Textbook Question

Explain why or why not

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


a.The sequence of partial sums for the series1 + 2 + 3 + ⋯ is {1, 3, 6, 10, …}.

Textbook Question

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


a. A series that converges must converge absolutely.

Textbook Question

72–75. {Use of Tech} Practical sequences

Consider the following situations that generate a sequence


a.Write out the first five terms of the sequence.


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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