Problem 122 Decide if the statements are tru... [FREE SOLUTION]

91影视

Calculus

Deborah Hughes-Hallett, Andrew M. Gleason, William G. McCallum

$Math Studyset 91影视 Explanations$ Math

7 Edition

Chapter 9: Problem 122

Decide if the statements are true or false. Give an explanation for your answer. If $b_{n} \leq a_{n} \leq 0$ for all $n$ and $\sum b_{n}$ converges, then $\sum a_{n}$ converges.

Short Answer

Expert verified

False, the convergence of $\sum b_n$ does not guarantee $\sum a_n$ converges.

Step by step solution

Understand the Given Inequality

The statement provides an inequality $b_{n} \leq a_{n} \leq 0$. This means each $b_n$ is less than or equal to $a_n$ and both are less than or equal to zero. Both $a_n$ and $b_n$ are non-positive sequences.

Analyze Convergence of $\sum b_n$

We are given that $\sum b_n$ converges. Since each $b_n \leq 0$, $\sum b_n$ is a convergent series of non-positive terms.

Apply the Comparison Test for Convergence

In the Comparison Test, if a series of non-positive terms $\sum b_n$ converges and $b_n \leq a_n \leq 0$, it does not necessarily imply $\sum a_n$ converges, because $a_n$ might still have larger magnitude as negative values.

Conclusion on the Convergence of $\sum a_n$

Given $b_n \leq a_n \leq 0$, we lack sufficient information about the absolute values of $a_n$ compared to $b_n$. Therefore, without additional criteria like $|a_n| \leq |b_n|$, $\sum a_n$ does not necessarily converge.

Unlock Step-by-Step Solutions & Ace Your Exams!

Full Textbook Solutions
Get detailed explanations and key concepts
Unlimited Al creation
Al flashcards, explanations, exams and more...
Ads-free access
To over 500 millions flashcards
Money-back guarantee
We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Comparison Test

The Comparison Test is a fundamental tool for determining the convergence of infinite series. It is particularly useful when dealing with non-positive or non-negative sequences. The main idea behind the test is to compare a known series with another series whose convergence properties are to be determined.

Let's break down how the Comparison Test works:

Given two series, $ \sum a_n $ and $ \sum b_n $, you can use the test under the condition that all terms in both series are non-positive or all are non-negative.
If $ 0 \leq b_n \leq a_n $ for all $ n $, and the series $ \sum a_n $ converges, then $ \sum b_n $ also converges.
Conversely, if $ a_n \leq b_n \leq 0 $, and the series $ \sum b_n $ converges, it does not automatically mean that $ \sum a_n $ converges if $ |a_n| $ can be larger than $ |b_n| $.

This highlights that for non-positive sequences, additional conditions are needed to ensure convergence beyond the knowledge that $ \sum b_n $ converges.

Non-positive Sequences

Non-positive sequences refer to sequences where each term is less than or equal to zero, i.e., each $ a_n \leq 0 $. These sequences are essential when analyzing series to understand their behavior, especially in the context of convergence.

Consider these key points about non-positive sequences:

Since all terms are non-positive, the series $ \sum a_n $ never increases, thus creating a non-decreasing partial sum when viewed in the context of absolute values.
When comparing two non-positive sequences, you must look at their absolute values when using tests for convergence, like the Comparison Test.
An important aspect of non-positive sequences is knowing that even if each term $ b_n $ in $ \sum b_n $ is less negative than $ a_n $ from $ \sum a_n $, the convergence isn't assured without more data.

Understanding these sequences allows better utilization of convergence tests and deeper insight into different series' behavior.

Convergent Series

A convergent series is one in which the sum of its infinitely many terms approaches a specific finite value as more terms are added. Identifying convergent series is a crucial part of mathematical analysis and is pivotal in many practical applications.

Key concepts for convergent series include:

For any series $ \sum a_n $, if the series has a finite limit as $ n \to \infty $, it is a convergent series.
Convergence doesn't depend solely on individual terms being small, but on their sum approaching a single value.
A series with negative terms, like with non-positive sequences, requires careful analysis through known tests (like the Comparison Test or Absolute Convergence Test) to ascertain convergence.

When dealing with assumptions based on convergence, such as $ \sum b_n $ being convergent, interpreting the conditions rigorously is crucial to drawing the correct conclusions about related series.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Recommended explanations on Math Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

91影视

Decide if the statements are true or false. Give an explanation for your answer. If \(b_{n} \leq a_{n} \leq 0\) for all \(n\) and \(\sum b_{n}\) converges, then \(\sum a_{n}\) converges.

Short Answer

Step by step solution

Understand the Given Inequality

Analyze Convergence of \(\sum b_n\)

Apply the Comparison Test for Convergence

Conclusion on the Convergence of \(\sum a_n\)

Key Concepts

Comparison Test

Non-positive Sequences

Convergent Series

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Applied Mathematics

Decision Maths

Calculus

Statistics

Mechanics Maths

Logic and Functions

Study anywhere. Anytime. Across all devices.