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Math Help - Pointwise limit

  1. #1
    Member thaopanda's Avatar
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    Uniformly Convergent

    Determine whether or not the sequence { f_{n}} _n \in N, where f_{n}(x) = n^2x^2e^{-nx} is uniformly convergent on:

    a.) [0,\infty)

    b.) (0,\infty)

    c.) [1,\infty)

    d.) And decide whether or not for f_{n}: [0,\infty) \rightarrow R if f_{n}(x) = \frac{nx}{1+n^2x^2} converges uniformly.
    Last edited by thaopanda; November 22nd 2009 at 03:22 PM. Reason: wrong title
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  2. #2
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    Opalg's Avatar
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    Quote Originally Posted by thaopanda View Post
    Determine whether or not the sequence { f_{n}} _n \in N, where f_{n}(x) = n^2x^2e^{-nx} is uniformly convergent on:

    a.) [0,\infty)

    b.) (0,\infty)

    c.) [1,\infty)

    d.) And decide whether or not for f_{n}: [0,\infty) \rightarrow R if f_{n}(x) = \frac{nx}{1+n^2x^2} converges uniformly.
    First, find the pointwise limit, which in this case is the zero function. To test whether or not f_n\to0, you need to investigate whether \sup\{|f(x)|\}\to0. The function is positive, so you don't need to worry about taking the absolute value, and you can find the maximum value of the function by calculus (find where the derivative is 0).
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