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Is this uniformly convergent?

  1. Jan 9, 2013 #1
    1. The problem statement, all variables and given/known data

    Consider [tex]f_n(x) = nx^n(1-x)[/tex] for x in [0,1].

    a) What is the limit of [tex]f_n(x)[/tex]?

    b) Does [tex]f_n \rightarrow f[/tex] uniformly on [0,1]?



    2. Relevant equations



    3. The attempt at a solution

    a) 0

    b) Yes...

    We know that [tex]sup|f_n(x) - f(x)| = |n{\frac{1}{2}}^n(1-\frac{1}{2})|[/tex]...

    and

    [tex]lim_{n \rightarrow \infty} [sup\{ |f_n(x) - f(x)|: x \in [0,1]\}] = 0[/tex]

    So it must be uniformly convergent on [0,1].

    Do you think my answer is correct?


    Thanks in advance
     
  2. jcsd
  3. Jan 9, 2013 #2

    micromass

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    Why? How do you know they will always obtain their maximum in 1/2??
     
  4. Jan 9, 2013 #3

    jbunniii

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    If we let [itex]y = 1-x[/itex], then we may write
    [tex]f_n(1-y) = n y (1 - y)^n[/tex]
    Now what happens if you choose [itex]y = 1/n[/itex]?
     
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