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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
Tutti i libri di testoBriggs 3rd EditionCh. 10 - Sequences and Infinite SeriesProblema 10.4.47b
Capitolo 10, Problema 10.4.47b

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.

Guida verificata passo dopo passo
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 constant.
Rewrite the given series \(\sum_{k=3}^{\infty} \frac{1}{\sqrt{k - 2}}\) by making a substitution to see if it matches the p-series form. Let \(j = k - 2\), so when \(k=3\), \(j=1\).
Express the series in terms of \(j\): \(\sum_{j=1}^{\infty} \frac{1}{\sqrt{j}} = \sum_{j=1}^{\infty} \frac{1}{j^{1/2}}\).
Since the series can be written as \(\sum_{j=1}^{\infty} \frac{1}{j^{1/2}}\), it matches the form of a p-series with \(p = \frac{1}{2}\).
Therefore, the given series is a p-series because it can be expressed in the form \(\sum \frac{1}{k^p}\) with \(p = \frac{1}{2}\).

Concetti chiave

Ecco i concetti essenziali che devi comprendere per rispondere correttamente alla domanda.

Definition of a p-series

A p-series is an infinite series of the form ∑ 1/n^p, where n starts from 1 or another positive integer, and p is a positive real number. The behavior and convergence of the series depend on the value of p.
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Percorso guidato
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P-Series and Harmonic Series

Index shift in series notation

Changing the index of summation or shifting the variable inside the series can transform the series into a more recognizable form. Understanding how to rewrite sums by adjusting indices helps identify the type of series.
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Geometric Series

Criteria for identifying p-series

To determine if a series is a p-series, the general term must be expressible as 1/(n^p) for some p. If the term involves a shifted index but can be rewritten to fit this form, it qualifies as a p-series.
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P-Series and Harmonic Series
Pratica correlata
Domanda del libro di testo

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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Loglog p-series Consider the series ∑ (k = 2 to ∞) 1 / (k(ln k)(ln ln k)ᵖ), where p is a real number.

b. Which of the following series converges faster? Explain.

∑ (k = 2 to ∞) 1 / (k(ln k)²) or ∑ (k = 3 to ∞) 1 / (k(ln k)(ln ln k)²)?

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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).

{-5, 5, -5, 5, ......}

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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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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.


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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