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Determine if series converges or diverges

  1. Apr 10, 2012 #1
    1. The problem statement, all variables and given/known data
    Ʃ cos^2(n)/(n^2+8)

    Ʃ 5n/(n^2+1) * cos(2πn)


    2. Relevant equations



    3. The attempt at a solution
    I think that both series diverge. Can anyone validate this or tell me if I'm wrong?
     
  2. jcsd
  3. Apr 10, 2012 #2
    Why do you think they diverge?
     
  4. Apr 10, 2012 #3
    I'm changing my mind. First one diverges because the cosine will oscillate between -1 and 1. The second one converges because it will always go to 0?
     
  5. Apr 10, 2012 #4
    Convergence and divergence of infinite series is a counterintuitive and complicated matter. Don't base you're conclusion of guesses and intuition. Use theorems instead. Do you know any? Which ones may be useful for those series?
     
  6. Apr 10, 2012 #5
    You have cos2n which will always be positive since it's squared. As a result, cos2n ≤ 1 and so cos2n/(n2 + 8) ≤ 1/(n2 + 8). From there it's not too difficult to show whether it converges or diverges using one of the series tests.

    Before looking at the second one, which series tests are you familiar with?
     
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