/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 59 Suppose that \(a_{n}>0\) and ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 10: Problem 59

Suppose that \(a_{n}>0\) and \(b_{n}>0\) for \(n \geq N(N\) an integer). If \(\lim _{n \rightarrow \infty}\left(a_{n} / b_{n}\right)=\infty\) and \(\sum a_{n}\) converges, can anything be said about \(\sum b_{n} ?\) Give reasons for your answer.

Short Answer

Expert verified
The series \(\sum b_n\) diverges.

Step by step solution

01

Understand the Problem Setup

We are given two sequences, \(a_n\) and \(b_n\), both greater than zero for sufficiently large \(n\). We know that the limit \(\lim _{n \rightarrow \infty} \frac{a_n}{b_n} = \infty\) and that the series \(\sum a_n\) converges. We need to determine what can be said about the convergence or divergence of the series \(\sum b_n\).
02

Analyze the Implications of the Limit

The condition \(\lim _{n \rightarrow \infty} \frac{a_n}{b_n} = \infty\) implies that \(\frac{a_n}{b_n}\) becomes arbitrarily large as \(n\) increases. This suggests that \(a_n\) is much larger than \(b_n\) for large \(n\). Mathematically, this means \(b_n\) is much smaller than \(a_n\) as \(n\) becomes large.
03

Apply Comparison Test for Series

Since \(a_n\) is much larger than \(b_n\), and \(\sum a_n\) converges, \(b_n\) is squeezed to be smaller than terms that converge absolutely. If \(a_n \geq M \cdot b_n\) for some positive constant \(M\) and sufficiently large \(n\), \(\sum b_n\) should converge by comparison because the terms are dominated by convergent series terms. However, \(\frac{a_n}{b_n}\) tends to infinity suggests \(b_n\) must be tending towards zero faster, but not sufficient enough to make the series \(\sum b_n\) converge by itself.
04

Draw Conclusion based on Series Behavior

Given that \(a_n\) tends to zero and that \(b_n\) does so even more rapidly, the series \(\sum b_n\) cannot converge. The rapid convergence of terms \(a_n\) assures that increasing values will not be canceled by \(b_n\). Thus, \(\sum b_n\) diverges as its corresponding sequence terms do not trend towards a limit necessary for convergence.

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 method used to determine whether a series converges or diverges by comparing it with another series. If you have a series \( \sum a_n \) and you know it converges, you can use another series \( \sum b_n \) to determine the same for it. For the Comparison Test:
  • If \( a_n \leq b_n \) for every \( n \) from some point on, and \( \sum b_n \) converges, then \( \sum a_n \) also converges.
  • Conversely, if \( a_n \geq b_n \) and \( \sum a_n \) diverges, \( \sum b_n \) also diverges.
In our exercise, although \( \sum a_n \) converges and \( \frac{a_n}{b_n} \to \infty \) implies \( a_n > b_n \) for large \( n \), \( \sum b_n \) is actually independent and must be examined more carefully as \( b_n \) is smaller, potentially leading to different behavior.
Convergence
Convergence of a series \( \sum a_n \) means that as you sum the terms, the total approaches a specific number. In formal terms, if the partial sums \( S_N = a_1 + a_2 + \cdots + a_N \) approach a finite limit as \( N \to \infty \), the series converges.This is crucial when evaluating a series like \( \sum a_n \). If we know it converges, it tells us the terms \( a_n \) themselves become very small, diminishing towards zero as \( n \) increases. This convergence information was not sufficient to conclude the same for \( \sum b_n \) as even with \( b_n \to 0 \), different factors in size and tendency can alter convergence behavior drastically.
Divergence
A series \( \sum b_n \) diverges if the partial sums \( S_N = b_1 + b_2 + \cdots + b_N \) do not approach a fixed number as \( N \to \infty \). This means that adding more terms keeps changing the sum's value infinitely. Divergence implies that the adding process continues without settling.In the exercise, the fact that \( \frac{a_n}{b_n} \rightarrow \infty \) and \( \sum a_n \) converges, points towards the possibility of \( \sum b_n \) diverging. This happens because although \( b_n \) becomes smaller as \( n \to \infty \), it might not decrease swiftly enough to counterbalance the endless addition of more terms.
Limit of a Sequence
Understanding the limit of a sequence is foundational when studying convergence and divergence. The sequence \( \{b_n\} \) is said to have a limit \( L \) as \( n \to \infty \) if \( b_n \to L \). This essentially means that the terms \( b_n \) get closer and closer to \( L \) as \( n \) increases.In our particular instance, since \( \frac{a_n}{b_n} \to \infty \), this indicates that \( a_n \) is larger relative to \( b_n \), making \( b_n \to 0 \) faster. However, the precise rate of \( b_n \) approaching zero can determine if the series \( \sum b_n \) will converge or diverge, leading to the conclusion that \( \sum b_n \) diverges due to insufficient decay rate.

One App. One Place for Learning.

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

Get started for free

Most popular questions from this chapter

Quadratic Approximations The Taylor polynomial of order 2 generated by a twice-differentiable function \(f(x)\) at \(x=a\) is called the quadratic approximation of \(f\) at \(x=a .\) Find the (a) linearization (Taylor polynomial of order 1 ) and (b) quadratic approximation of \(f\) at \(x=0\). \(f(x)=\tan x\)

Quadratic Approximations The Taylor polynomial of order 2 generated by a twice-differentiable function \(f(x)\) at \(x=a\) is called the quadratic approximation of \(f\) at \(x=a .\) Find the (a) linearization (Taylor polynomial of order 1 ) and (b) quadratic approximation of \(f\) at \(x=0\). \(f(x)=\cosh x\)

Which of the sequences converge, and which diverge? Give reasons for your answers. $$ a_{n}=1-\frac{1}{n} $$

Which of the series in Exercises 13 46 converge, and which diverge? Give reasons for your answers. (When you check an answer, remember that there may be more than one way to determine the series' convergence or divergence.) $$ \sum_{n=1}^{\infty} n \tan \frac{1}{n} $$

Which of the sequences converge, and which diverge? Give reasons for your answers. $$ a_{n}=\frac{2^{n}-1}{2^{n}} $$
See all solutions

Recommended explanations on Math Textbooks

Mechanics Maths

Read Explanation

Pure Maths

Read Explanation

Probability and Statistics

Read Explanation

Discrete Mathematics

Read Explanation

Calculus

Read Explanation

Statistics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.