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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 10.4.47b
Chapter 10, Problem 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.

Verified step by step guidance
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 \(p = \frac{1}{2}\).

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 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.
Recommended video:
04:30
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.
Recommended video:
06:00
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.
Recommended video:
04:30
P-Series and Harmonic Series
Related Practice
Textbook Question

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


b. (2n)! / (2n − 1)! = 2n

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Textbook Question

Explain why or why not

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


b.If a sequence of positive numbers converges, then the sequence is decreasing.

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Textbook Question

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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Textbook Question

{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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Textbook Question

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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Textbook Question

{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?

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