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Math Help - Pointwise and uniform convergence

  1. #1
    Member thaopanda's Avatar
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    Pointwise and uniform convergence

    a) For each n \in N consider F_{n} : [0, \infty) \rightarrow R be given by F_{n}(x) := (x^n)e^{-nx}. Show that { F_{n}} _{n \in N} converges in a pointwise fashion to the zero function on [0, \infty).

    b) Decide whether the convergence in part a is uniform on [0, \infty).

    please help
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  2. #2
    Super Member redsoxfan325's Avatar
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    Quote Originally Posted by thaopanda View Post
    a) For each n \in N consider F_{n} : [0, \infty) \rightarrow R be given by F_{n}(x) := (x^n)e^{-nx}. Show that { F_{n}} _{n \in N} converges in a pointwise fashion to the zero function on [0, \infty).

    b) Decide whether the convergence in part a is uniform on [0, \infty).

    please help
    a) Well, for all x, \lim_{n\to\infty}\frac{x^n}{e^{nx}}=0. One way to see this is that if you use L'Hopital's Rule n times, you'll get \lim_{n\to\infty}\frac{n!}{n!e^{nx}}=0

    b) The derivative of x^ne^{-nx} is nx^{n-1}e^{-nx}(1-x), so the max on [0,\infty) occurs at x=1 (for all n), and it follows that x^ne^{-nx}\leq e^{-n}, which can clearly be made less than any \epsilon. (Choosing n>-\ln\epsilon does it.)
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