Download the App!

Get 24/7 study help with the Numerade app for iOS and Android! Enter your email for an invite.

Sent to:
Search glass icon
  • Login
  • Textbooks
  • Ask our Educators
  • Study Tools
    Study Groups Bootcamps Quizzes AI Tutor iOS Student App Android Student App StudyParty
  • For Educators
    Become an educator Educator app for iPad Our educators
  • For Schools

Question

Answered step-by-step

Problem 46 Hard Difficulty

If $ \sum a_n $ and $ \sum b_n $ are both convergent series with positive terms, is it true that $ \sum a_n b_n $ is also convergent?


Video Answer

Solved by verified expert

preview
Numerade Logo

This problem has been solved!

Try Numerade free for 7 days

JH
J Hardin
Numerade Educator

Like

Report

Textbook Answer

Official textbook answer

Video by J Hardin

Numerade Educator

This textbook answer is only visible when subscribed! Please subscribe to view the answer

Related Courses

Calculus 2 / BC

Calculus: Early Transcendentals

Chapter 11

Infinite Sequences and Series

Section 4

The Comparison Tests

Related Topics

Sequences

Series

Discussion

You must be signed in to discuss.
Top Calculus 2 / BC Educators
Anna Marie Vagnozzi

Campbell University

Kayleah Tsai

Harvey Mudd College

Caleb Elmore

Baylor University

Michael Jacobsen

Idaho State University

Calculus 2 / BC Courses

Lectures

Video Thumbnail

01:59

Series - Intro

In mathematics, a series is, informally speaking, the sum of the terms of an infinite sequence. The sum of a finite sequence of real numbers is called a finite series. The sum of an infinite sequence of real numbers may or may not have a well-defined sum, and may or may not be equal to the limit of the sequence, if it exists. The study of the sums of infinite sequences is a major area in mathematics known as analysis.

Video Thumbnail

02:28

Sequences - Intro

In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed. Like a set, it contains members (also called elements, or terms). The number of elements (possibly infinite) is called the length of the sequence. Unlike a set, order matters, and exactly the same elements can appear multiple times at different positions in the sequence. Formally, a sequence can be defined as a function whose domain is either the set of the natural numbers (for infinite sequences) or the set of the first "n" natural numbers (for a finite sequence). A sequence can be thought of as a list of elements with a particular order. Sequences are useful in a number of mathematical disciplines for studying functions, spaces, and other mathematical structures using the convergence properties of sequences. In particular, sequences are the basis for series, which are important in differential equations and analysis. Sequences are also of interest in their own right and can be studied as patterns or puzzles, such as in the study of prime numbers.

Join Course
Recommended Videos

04:40

If $ \sum a_n $ is a conve…

02:29

If $\Sigma a_{n}$ and $\Si…

00:56

If $\sum a_{n}$ and $\sum …

01:54

If $\Sigma a_{n}$ is a con…

Watch More Solved Questions in Chapter 11

Problem 1
Problem 2
Problem 3
Problem 4
Problem 5
Problem 6
Problem 7
Problem 8
Problem 9
Problem 10
Problem 11
Problem 12
Problem 13
Problem 14
Problem 15
Problem 16
Problem 17
Problem 18
Problem 19
Problem 20
Problem 21
Problem 22
Problem 23
Problem 24
Problem 25
Problem 26
Problem 27
Problem 28
Problem 29
Problem 30
Problem 31
Problem 32
Problem 33
Problem 34
Problem 35
Problem 36
Problem 37
Problem 38
Problem 39
Problem 40
Problem 41
Problem 42
Problem 43
Problem 44
Problem 45
Problem 46

Video Transcript

if the's sums are both conversion and we have dealing with positive terms, so this means and and be enter both positive. Is it true that this some also convergence? So let's try to see if this is true here. So fence, there's some converges by the diversions test. This is from section eleven point two. So let me write that on the side eleven point two. We have that be and must approach zero in the limit. Since B ends eventually are getting very close to zero. There exist and inside your end. So let's say a natural number and such that if we let the index little and be larger than that, then the fiend must be less than or equal to, Let's say, one. This is just a consequence of the limit being zero sense it zero. That means there's a point at which B n plus one B and plus two and so on. All these numbers will be less than or equal to one. So all I'm saying here, therefore, because of this fact, I can rewrite this some let's say there are using a starting point here unless you say the starting point is one. It doesn't really matter what the starting point is. You could replace that one with any other number, and this will still be true. So first, let me instead of writing is infinite. Some break this into two parts. So look at the sun for all the way up to Capitol. And then I'll look at the remaining part now in this some over here, our little and is bigger than capital and therefore, for these values and this sum over here in the green, I can go ahead and use the fact that being is less than or equal to one due to this fact. So that means that our Siri's over here is less than or equal to the sum from one to capitol and a NBN. But then over here I'm just replacing being with one, and I have the inequality here. So this inequality is what made me write this inequality here. And I just have the sum from capital and plus one to infinity of a end. All I'm doing here is a NBN less than or equal to a M times one equals and and finally I know that this some here let me use a different color this red some over here, which I'm now using blue. This some converges because we were given that there's some over here convergence and the sum from N plus one to infinity of a end is less than or equal to, actually strictly less than the sum from one toe infinity of AM so sense we're given at this one emerges then by Red, a comparison test. This also converges. So let's say, by comparison with the entire series of the A M so sense it converges and rolling, adding positive numbers. That just means that we just showed that the sum from N plus one to infinity of a M is less than infinity. However, this original sum over here in the green, which I'm not circling this is automatically less than infinity because it's only a finite sum. Therefore, we're adding two numbers together that are less than infinity, So the result will also be lesson infinity. So therefore, we have shown that the sum of the A m. Bien first of all, we know it has to be larger than zero because a end and beyond are positive. On the other hand, We just showed that it's less than infinity, so it has to converge to a real number because again we're only dealing with positive terms, and that's our final answer.

Get More Help with this Textbook
James Stewart

Calculus: Early Transcendentals

View More Answers From This Book

Find Another Textbook

Study Groups
Study with other students and unlock Numerade solutions for free.
Math (Geometry, Algebra I and II) with Nancy
Arrow icon
Participants icon
96
Hosted by: Ay?Enur Çal???R
Math (Algebra 2 & AP Calculus AB) with Yovanny
Arrow icon
Participants icon
55
Hosted by: Alonso M
See More

Related Topics

Sequences

Series

Top Calculus 2 / BC Educators
Anna Marie Vagnozzi

Campbell University

Kayleah Tsai

Harvey Mudd College

Caleb Elmore

Baylor University

Michael Jacobsen

Idaho State University

Calculus 2 / BC Courses

Lectures

Video Thumbnail

01:59

Series - Intro

In mathematics, a series is, informally speaking, the sum of the terms of an infinite sequence. The sum of a finite sequence of real numbers is called a finite series. The sum of an infinite sequence of real numbers may or may not have a well-defined sum, and may or may not be equal to the limit of the sequence, if it exists. The study of the sums of infinite sequences is a major area in mathematics known as analysis.

Video Thumbnail

02:28

Sequences - Intro

In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed. Like a set, it contains members (also called elements, or terms). The number of elements (possibly infinite) is called the length of the sequence. Unlike a set, order matters, and exactly the same elements can appear multiple times at different positions in the sequence. Formally, a sequence can be defined as a function whose domain is either the set of the natural numbers (for infinite sequences) or the set of the first "n" natural numbers (for a finite sequence). A sequence can be thought of as a list of elements with a particular order. Sequences are useful in a number of mathematical disciplines for studying functions, spaces, and other mathematical structures using the convergence properties of sequences. In particular, sequences are the basis for series, which are important in differential equations and analysis. Sequences are also of interest in their own right and can be studied as patterns or puzzles, such as in the study of prime numbers.

Join Course
Recommended Videos

04:40

If $ \sum a_n $ is a convergent series with positive terms, is it true that $ \…

02:29

If $\Sigma a_{n}$ and $\Sigma b_{n}$ are both convergent series with positive t…

00:56

If $\sum a_{n}$ and $\sum b_{n}$ are both convergent series, is $\sum a_{n} b_{…

01:54

If $\Sigma a_{n}$ is a convergent series with positive terms, is it true that $…

Add To Playlist

Hmmm, doesn't seem like you have any playlists. Please add your first playlist.

Create a New Playlist

`

Share Question

Copy Link

OR

Enter Friends' Emails

Report Question

Get 24/7 study help with our app

 

Available on iOS and Android

About
  • Our Story
  • Careers
  • Our Educators
  • Numerade Blog
Browse
  • Bootcamps
  • Books
  • Notes & Exams NEW
  • Topics
  • Test Prep
  • Ask Directory
  • Online Tutors
  • Tutors Near Me
Support
  • Help
  • Privacy Policy
  • Terms of Service
Get started