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Rate of convergeance

  1. Oct 3, 2012 #1
    Ʃ([itex]\frac{1}{n}[/itex]) diverges but Ʃ([itex]\frac{1}{n^2}[/itex]) converges but both corresponding sequences converge (I think). So at what rate does a sequence have to converge for the corresponding series to converge.

    Or asked differently what is the largest value x can be for Ʃ([itex]\frac{1}{n^x}[/itex]) to diverge
  2. jcsd
  3. Oct 3, 2012 #2


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    The dividing line is at p = 1. ##\sum \frac 1 {x^p}## converges if ##p>1##, as you can verify by the integral test.
  4. Oct 3, 2012 #3
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