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Math Help - limit function / uniform convergence

  1. #1
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    limit function / uniform convergence

    Examine the functions f_n(x) = x^(2n) / (1+x^(2n)) on R for n = 1,2,...
    a) calculate the limit function f of the sequence [f_n]
    b) Complete the following sentence: If [f_n] converges uniformly on the interval I=[a,b], then I cannot contain the point(s) x=____. Justify your answer, and then show the convergence is uniform on any closed intervals not containing those points.
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  2. #2
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    The function f_{n}(x) can also be written as \displaystyle f_{n}(x)=1- \frac{1}{1+x^{2n}} . From here we see that \lim_{n \to \infty} f_{n}(x)=1 if  x \neq 0 and \lim_{n \to \infty} f_{n}(x)=0 \ \textrm{if x=0}. As for the next part i think that I cannot contain x<0
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  3. #3
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    Quote Originally Posted by bmixon View Post
    Examine the functions f_n(x) = x^(2n) / (1+x^(2n)) on R for n = 1,2,...
    a) calculate the limit function f of the sequence [f_n]
    b) Complete the following sentence: If [f_n] converges uniformly on the interval I=[a,b], then I cannot contain the point(s) x=____. Justify your answer, and then show the convergence is uniform on any closed intervals not containing those points.
    If |x|<1 the function goes to 0. If |x|>1 the function goes to 1. f(1) =
    f(-1)=1/2. Thus, the convergence cant be uniform in intervals contaiining 1 or -1.
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