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Math Help - uniform convergence

  1. #1
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    uniform convergence

    (a) Fn(x) = nxe^(-nx), x>= 0. Prove that the convergence is uniform for x>= @, where @ is any positive number. Why is the convergence not uniform on the interval 0<= x<= @??

    (b) Fn(x) = (tan^-1)nx, all x. What conditions must a and b satisfy if the convergence is to be uniform on a <= x <= b ?
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  2. #2
    MHF Contributor FernandoRevilla's Avatar
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    Quote Originally Posted by mathsohard View Post
    (a) Fn(x) = nxe^(-nx), x>= 0. Prove that the convergence is uniform for x>= @, where @ is any positive number. Why is the convergence not uniform on the interval 0<= x<= @??

    For x\in [0,+\infty) the function f_n(x)=nxe^{-nx} has an absolute maximum at x=1/n so, if n\geq 1/a then, f_n is decreasing. For n\geq 1/a and x\geq a we have 0\leq f_n(x)\leq f_n(a) .

    But \lim_{n\to +\infty}f_n(a)=0 so, f_n\to 0 in [a,+\infty) uniformly .

    Try the other part.


    Fernando Revilla
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