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Deceptive uniform convergence question

  1. Mar 31, 2013 #1


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    1. The problem statement, all variables and given/known data


    2. Relevant equations

    Theorem III :

    3. The attempt at a solution

    At first this question had me jumping to a wrong conclusion.

    Upon closer inspection I see the sequence converges to 1 as n goes to infinity for |x|<1. The sequence converges to 0 as n goes to infinity for |x|≥1. Hence the sequence is not uniformly convergent over the whole real line.

    If we restrict the domain of x to (-1,1) or (-∞,-1] U [1,∞), then we can observe uniform convergence over each interval respectively.

    The question isn't too clear about what it's asking for, but that's my take.
  2. jcsd
  3. Mar 31, 2013 #2


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    I think you are supposed to conclude that given the theorem, if the ##f_n## are continuous and the limit function is not continuous, then the convergence can't be uniform. And I don't think they converge uniformly on (-1,1) either or any of the other intervals you are talking about. If you think they do please let me know why.
  4. Apr 1, 2013 #3


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    I see, I think i understand how the theorem and the question relate. I never did check the convergence on the intervals though so I suppose I shouldn't have assumed.
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