Calculus 2 interval of convergence -- checking end points

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The discussion focuses on finding the interval of convergence for a series and determining the convergence or divergence at the endpoints. The original poster has successfully identified the interval but struggles with the appropriate tests for the endpoints. Participants emphasize the importance of using the ratio test and suggest that if x is negative, the series may be alternating. There is a request for clearer presentation of the problem, as the initial image was unreadable, prompting the poster to provide a new link. The conversation highlights the need for clarity in mathematical communication to facilitate effective assistance.
rossmoesis
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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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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:
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.
 
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.
 

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