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Quickie Ratio Test question

  1. Dec 11, 2005 #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?
  2. jcsd
  3. Dec 11, 2005 #2


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    You could use the alternating series test.

    More simply, you can relate the series at -1/3 to the series at 1/3...
  4. Dec 11, 2005 #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?
  5. Dec 11, 2005 #4


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    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.
  6. Dec 11, 2005 #5
    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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