Need help with a series (radius, convergence)

Click For Summary

Homework Help Overview

The problem involves determining the radius and interval of convergence for a given series expressed as the sum of (n*x^n)/((4^n)((n^2) +1)). The context is centered around series convergence, specifically absolute and conditional convergence.

Discussion Character

  • Exploratory, Mathematical reasoning, Assumption checking

Approaches and Questions Raised

  • Participants discuss starting with the ratio test to analyze the series. There are varying interpretations of the limit obtained during the test, with some questioning the results and suggesting a need for verification. Others propose checking the endpoints of the interval after determining the radius.

Discussion Status

The discussion is active, with participants exploring different approaches to applying the ratio test. Some have provided guidance on how to set up inequalities based on the limit found, while others emphasize the importance of checking convergence at the endpoints. There is no explicit consensus on the final radius or interval yet.

Contextual Notes

Participants note potential discrepancies in the limit results and the need for careful checking of boundary conditions, as the ratio test may not apply at certain points. The original poster expresses uncertainty about the initial steps in solving the problem.

dmitriylm
Messages
39
Reaction score
2

Homework Statement


Find the series' radius and interval of convergence. What what value of x does the series converge absolutely, conditionally?

Sum (n=0 to infinity) (nx^n)/((4^n)((n^2) +1)))


Homework Equations





The Attempt at a Solution


Not quite sure where to start with this. I've been having trouble with series in this course because its not as straightforward as all the other problems.
 
Physics news on Phys.org
haven't tried it, but would start with a ratio test
 
I was taught to start these problems with the ratio test. Then when you take the limit as n approaches infinity you can factor out the x. If the limit is 0 the radius is infinity and if the limit is infinity the radius is 0. If it it a number (c) you can multiply it by the x you factored out and set up the inequality -1<CX<1. Then solve for x and find if it converges at the endpoints by putting them in the initial sum and using any convergence test you wish, this will give you the radius of convergence.
 
Doing the ratio test and checking for the limit I found the limit to be x/4. Where do I go from there?
 
-1<x/4<1 so to solve for x multiply everything by 4 which gives -4<x<4 so the interval is -4,4 but we don't know if it is a closed or opened interval. To determine this plug -4 in for x in the original series and find if it converges or diverges if it converges it will be a bracket on the -4 if it diverges it is a parenthesis then do the same for 4.
 
however i tried it real quick and got 4x not x/4. I could be wrong but it might be worth a double check.
 
and you will nee dto check the boundary individually as the ration test does not apply for a ratio of 1
 
DCASH88 said:
however i tried it real quick and got 4x not x/4. I could be wrong but it might be worth a double check.
No, you are correct. The radius of convergence is 1/4, not 4.

The series will converge absolutely inside the radius of converge, -1/4< x< 1/4. It may converge absolutely, conditionally, or not converge at x= -1/4 and/or x=1/4. Those will have to be checked separately.
 

Similar threads

On the definition of radius of convergence; a small supremum technicality
  • · Replies 7 ·
Replies
7
Views
2K
Valid conclusion for an absolutely convergent sequence
  • · Replies 4 ·
Replies
4
Views
1K
Studying the convergence of a series with an arctangent of a partial sum
  • · Replies 7 ·
Replies
7
Views
2K
How to show that these sequences are summable?
  • · Replies 1 ·
Replies
1
Views
2K
Intervals of Convergence- Power Series
  • · Replies 11 ·
Replies
11
Views
3K
Addition of power series and radius of convergence
  • · Replies 14 ·
Replies
14
Views
2K
Interval of uniform convergence of a series
  • · Replies 24 ·
Replies
24
Views
3K
Set of convergence of a Power series
  • · Replies 4 ·
Replies
4
Views
2K
Study the convergence and absolute convergence of the following series
  • · Replies 5 ·
Replies
5
Views
2K
Radius of Convergence and Power Series Expansion Around Multiple Values
  • · Replies 6 ·
Replies
6
Views
2K