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Convergent uniformly function

  1. Nov 5, 2012 #1
    1. The problem statement, all variables and given/known data
    define function fn:[0,1]->R by fn(x)=(n^p)x*exp(-(n^q)x) where p , q>0 and
    fn->0 pointwise on[0,1] as n->infinite.,with the pointwise limit f(x)=0 ,and sup|fn(x)|=(n^(p-q))/e
    assume that ε is in (0,1)
    does fn converges uniformly on [1,1-ε]? how about on[0.1-ε]????

    3. The attempt at a solution
    my idea is checking whether the pointwise limit f is continuous on the interval above,but it is obvious,then f is continuous ,so fn is uniformly convergent,but i thought it is wrong. can someone give me any idea?????
     
  2. jcsd
  3. Nov 5, 2012 #2

    SammyS

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    First of all, that looks like a typo → does fn converge uniformly ... how about on[0.1-ε]. Shouldn't that be [0,1-ε] ?

    As for the problem:

    The answer to this question may depend upon the values of p & q, or at least upon the value of p-q .

    Consider the supremum: of fn which is: [itex]\displaystyle\frac{n^{p-q}}{e}\ .[/itex]

    At what value of x, does the supremum occur?
     
  4. Nov 6, 2012 #3

    yes, i know that the supremum occur at 1/(n^q),which is always smaller or equal to 1, but whats the relation for this x value and the interval. i had proved that this function if p<q it is convergent uniformly on [0,1] ,if p>=q it is not uniformly convergent on [0,1] because the supremum goes to infinite. but i cannot see whats the different when i change the intevrval to [0,1-ε]..
     
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