Calculus 2 interval of convergence -- checking end points

You have one positive term and one negative term, and the absolute values of the terms are decreasing as n increases. That's the Leibniz test, and that would tell you that the series converges.The only way to make sure that your end points converge is to test them. The best way to test them is to use the alternating series test. So, you want to show that if x = 3, then the series converges and that if x = -3, then the series converges. You can do this by comparing the absolute values of the terms in your series to the absolute values of the terms in two other series. One of the two other series should be a geometric series. What should the other series be
  • #1
rossmoesis
6
0

Homework Statement


[PLAIN]http://email.photobucket.com/wf/click?upn=tRCgAMfEfk8moxY0TjPC4lPPM0sXOKwutyIq47CMTKs0dzyywVYzTaMWH8-2Bzo5-2FsAT1hSn2u8nhtTlXRVI-2BDs3dbLwqV7FVPJZdCfwS6qDorV0c-2Blq6XRnrVUYsLffi2id9Ma4XDGYssMtivke9T0A-3D-3D_whPbjyDNqQqG-2FABCOAcFDhlciHlWVosDuzGNcx8ulLn1XPsw3RItB9a0ZKzbWUjhD5QN0BjKqPT3p-2FFMYnHwCHkxJ8U7UgTdccAw-2BQNDt-2FWYcVOcGYWyPrDnhFZ4wxeKwtHSMH-2FU0Ev-2FD82L-2F5FVMYpPr42BkMfKA0c7B9UFHMZbfml3RpIMxX73T-2BjugVi6TYP6-2BFgjtfKZemv5vG8xCDZrA74p9b7HQm1E0kHNLeQDGCyjMm4HzzVR9GA23nwt
http://email.photobucket.com/wf/click?upn=tRCgAMfEfk8moxY0TjPC4lPPM0sXOKwutyIq47CMTKs0dzyywVYzTaMWH8-2Bzo5-2FsAT1hSn2u8nhtTlXRVI-2BDs3dbLwqV7FVPJZdCfwS6qDorV0c-2Blq6XRnrVUYsLffi2id9Ma4XDGYssMtivke9T0A-3D-3D_whPbjyDNqQqG-2FABCOAcFDhlciHlWVosDuzGNcx8ulLn1XPsw3RItB9a0ZKzbWUjhD5QN0BjKqPT3p-2FFMYnHwCHkxJ8U7UgTdccAw-2BQNDt-2FWYcVOcGYWyPrDnhFZ4wxeKwtHSMH-2FU0Ev-2FD82L-2F5FVMYpPr42BkMfKA0c7B9UFHMZbfml3RpIMxX73T-2BjugVi6TYP6-2BFgjtfKZemv5vG8xCDZrA74p9b7HQm1E0kHNLeQDGCyjMm4HzzVR9GA23nwt

Homework Equations



Ratio test

The Attempt at a Solution



I found the interval of convergence. I'm having a hard time finding the correct test to make sure my end points either converge or diverge. I attached a picture of my work. Some hints would be greatly appreciated.
 
Last edited by a moderator:
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  • #2
rossmoesis said:

Homework Statement


http://email.photobucket.com/wf/click?upn=tRCgAMfEfk8moxY0TjPC4lPPM0sXOKwutyIq47CMTKs0dzyywVYzTaMWH8-2Bzo5-2FsAT1hSn2u8nhtTlXRVI-2BDs3dbLwqV7FVPJZdCfwS6qDorV0c-2Blq6XRnrVUYsLffi2id9Ma4XDGYssMtivke9T0A-3D-3D_whPbjyDNqQqG-2FABCOAcFDhlciHlWVosDuzGNcx8ulLn1XPsw3RItB9a0ZKzbWUjhD5QN0BjKqPT3p-2FFMYnHwCHkxJ8U7UgTdccAw-2BQNDt-2FWYcVOcGYWyPrDnhFZ4wxeKwtHSMH-2FU0Ev-2FD82L-2F5FVMYpPr42BkMfKA0c7B9UFHMZbfml3RpIMxX73T-2BjugVi6TYP6-2BFgjtfKZemv5vG8xCDZrA74p9b7HQm1E0kHNLeQDGCyjMm4HzzVR9GA23nwt
http://email.photobucket.com/wf/click?upn=tRCgAMfEfk8moxY0TjPC4lPPM0sXOKwutyIq47CMTKs0dzyywVYzTaMWH8-2Bzo5-2FsAT1hSn2u8nhtTlXRVI-2BDs3dbLwqV7FVPJZdCfwS6qDorV0c-2Blq6XRnrVUYsLffi2id9Ma4XDGYssMtivke9T0A-3D-3D_whPbjyDNqQqG-2FABCOAcFDhlciHlWVosDuzGNcx8ulLn1XPsw3RItB9a0ZKzbWUjhD5QN0BjKqPT3p-2FFMYnHwCHkxJ8U7UgTdccAw-2BQNDt-2FWYcVOcGYWyPrDnhFZ4wxeKwtHSMH-2FU0Ev-2FD82L-2F5FVMYpPr42BkMfKA0c7B9UFHMZbfml3RpIMxX73T-2BjugVi6TYP6-2BFgjtfKZemv5vG8xCDZrA74p9b7HQm1E0kHNLeQDGCyjMm4HzzVR9GA23nwt

Homework Equations



Ratio test

The Attempt at a Solution



I found the interval of convergence. I'm having a hard time finding the correct test to make sure my end points either converge or diverge. I attached a picture of my work. Some hints would be greatly appreciated.

Your photo is unreadable on my screen. Please type it all out (as per PF standards), but if you still insist on posting an image, at least work at making it usable.
 
Last edited by a moderator:
  • #3
Ray Vickson said:
Your photo is unreadable on my screen. Please type it all out (as per PF standards), but if you still insist on posting an image, at least work at making it usable.

I tried to go back and edit the post but was unable to. This link should work for the problem

http://s30.postimg.org/k3ksocech/scan.jpg

thanks for the heads up.
 
  • #4
rossmoesis said:
I tried to go back and edit the post but was unable to. This link should work for the problem

http://s30.postimg.org/k3ksocech/scan.jpg

thanks for the heads up.

If x is negative, then you have an alternating series.
 

1. What is the interval of convergence for a power series?

The interval of convergence for a power series is the range of values for which the series converges. In other words, it is the set of all x-values for which the series will produce a finite value.

2. How do you check the end points of an interval of convergence?

To check the end points of an interval of convergence, you can use either the Ratio Test or the Root Test. These tests will determine if the series converges or diverges at the end points of the interval.

3. What happens if the end points of an interval of convergence are included?

If the end points of an interval of convergence are included, it means that the series may or may not converge at those points. Additional tests, such as the Limit Comparison Test, can be used to determine the convergence or divergence at the end points.

4. Can the interval of convergence be different for different values of x?

Yes, the interval of convergence can vary for different values of x. This is because the convergence of a power series depends on the x-values being plugged into the series. Different values of x can result in different intervals of convergence.

5. How can I use the interval of convergence to find the sum of a power series?

The interval of convergence can be used to find the sum of a power series by plugging in the x-values within the interval into the series. This will give the value of the series at those points. However, it is important to note that the series may or may not converge at the end points of the interval, so additional tests may be needed to determine the sum at those points.

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