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

Problem

Suppose that the radius of convergence of the pow…

01:49

Question

Answered step-by-step

Problem 41 Medium Difficulty

Suppose the series $ \sum c_n x^n $ has radius of convergence 2 and the series $ \sum d_n x^n $ has radius of convergence 3. What is the radius of convergence of the series $ \sum (c_n + d_n) x^n? $


Video Answer

Solved by verified expert

preview
Numerade Logo

This problem has been solved!

Try Numerade free for 7 days

Gabriel Rhodes
Numerade Educator

Like

Report

Textbook Answer

Official textbook answer

Video by Gabriel Rhodes

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 8

Power Series

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

Samuel Hannah

University of Nottingham

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

01:38

Suppose the series $\Sigma…

01:49

Suppose that the radius of…

02:41

Suppose that the radius of…

02:16

Suppose the series $\Sigma…

04:18

Suppose that the radius of…

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

Video Transcript

since this sum here has a larger radius of convergence and this sum, we know that these dian terms we're going to be approaching zero significantly faster than these see in terms. So much so that as n goes to infinity, Deanne overseeing is going to go to zero. Now we can use the fact that our radius of convergence is the limit as n goes to infinity of absolute value of C N plus stian over C n plus one plus D in class one. And now we can divide both sides both the top and the bottom by CNN to get one plus D and overseeing over C N plus one oversea end plus the end plus one overseeing as we mentioned, since these dian terms go to zero much faster than these see in terms, this is going to go to zero. This is going to go to zero and then we'LL just have one divided by C n plus one over seeing plus one. So division is the same thing as multiplying by the reciprocal and so we get here. Okay, but this should be the radius of convergence for this sum here, which we set is too. Okay. So in fact, whenever you have a sum, some power Siri's and some other power Siri's and you're adding them up in this way, that radius of convergence for the sum is just gonna be the smaller of the two unless they happen to have the same radius of convergence, In which case, there's not quite as much to say.

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
83
Hosted by: Ay?Enur Çal???R
Math (Algebra 2 & AP Calculus AB) with Yovanny
Arrow icon
Participants icon
52
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

Samuel Hannah

University of Nottingham

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

01:38

Suppose the series $\Sigma c_{n} x^{n}$ has radius of convergence 2 and the ser…

01:49

Suppose that the radius of convergence of the power series $ \sum c_n x^n $ is …

02:41

Suppose that the radius of convergence of the power series $\sum c_{n} x^{n}$ i…

02:16

Suppose the series $\Sigma c_{n} x^{n}$ has radius of convergence 2 and the se…

04:18

Suppose that the radius of convergence of the power series $\Sigma c_{n} x^{n}$…
Additional Mathematics Questions

01:58

Graphing Linear Inequality in Two Variables
Inatructions: Using a graphin…

02:48

Find the perimeter of AUVW . Round your answer to the nearest tenth if neces…

04:23

23 - 24. Given the figure below, the relationships among chords, arcs, centr…

02:56

A B and €C are three similar solids
The surface area of A is 24cm? The su…

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