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A basic convergence question

  1. Nov 25, 2006 #1
    How would I show that the series whose terms are given by
    (cos n)/(1+n) does not converge absolutely? Thanks so much!
  2. jcsd
  3. Nov 25, 2006 #2
    [tex] \sum_{n=0}^{\infty} \frac{\cos n}{1+n} [/tex]

    So [tex] \sum_{n=0}^{\infty} \frac{\cos n}{1+n} \sim \frac{1}{n} [/tex]
    Last edited: Nov 25, 2006
  4. Nov 25, 2006 #3
    Hi sorry, I'm having trouble understanding that. How am I supposed to compare that?
  5. Nov 25, 2006 #4
    Is it cos(n) or cos(n*pi)? If it is the first then the limit comparison test should wor fairly well with the series 1/n.
  6. Nov 26, 2006 #5
    You mean this limit comparison test? http://mathworld.wolfram.com/LimitComparisonTest.html

    But what limit does it tend to? I thought |cos n| didn't tend to a limit as n--> infinity.
  7. Nov 26, 2006 #6
    Yea that test, cos(n) doesn't but it is bounded so I think if you use that fact and maybe the squeeze theorem you should be able to show that the series doesn't converge absolutely. It shouldn't be very hard to show that the series does converge as is using the alternating series test, but I'm not sure if it is cos(n) as opposed to cos(pi*n).
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