Need help with a series (radius, convergence)

[dmitriylm]

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.

 

[lanedance]

haven't tried it, but would start with a ratio test

 

[DCASH88]

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.

 

[dmitriylm]

Doing the ratio test and checking for the limit I found the limit to be x/4. Where do I go from there?

 

[DCASH88]

-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.

 

[DCASH88]

however i tried it real quick and got 4x not x/4. I could be wrong but it might be worth a double check.

 

[lanedance]

and you will nee dto check the boundary individually as teh ration test does not apply for a ratio of 1

 

[HallsofIvy]

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.