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Quickie Ratio Test question

  • #1
Have to find the radius of convergence and interval of convergence,
the series is (3^n x^n ) / (n+1)^2,
did the ratio test and found the radius of convergence to be the 1/3.
now for finding the interval of convergence I plug in -1/3 and 1/3 into x and find out if it converges or not

For 1/3, it converges due to p-series, 2>1.

But for -1/3 I know it converges but can see why? Any help here at this endpoint?
 

Answers and Replies

  • #2
shmoe
Science Advisor
Homework Helper
1,992
1
You could use the alternating series test.

More simply, you can relate the series at -1/3 to the series at 1/3...
 
  • #3
I dont quite understand, the alternating series only works when An+1< or = to An, and in that series it doesnt, because An is negative and An+1 is postivie?
Could you elaborate and how I would relate it to the series at 1/3?
 
  • #4
shmoe
Science Advisor
Homework Helper
1,992
1
Look at the alternating series test again, it's the absolute values of the terms that are decreasing (and going to zero) while the sign is alternating.


The series at 1/3 is the absolute values of the terms of the series at -1/3, i.e. you've already should that the series at -1/3 is absolutely convergent.
 
  • #5
iiiiiiiiiii...
my book doesnt say absolute value... so idk. I see what you mean if thats true. Why doesnt my book say that it says just that its decreasing or equal too...
 

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