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Positive real numbers question

  1. Nov 8, 2008 #1
    question:
    let [tex]x_1,...,x_n[/tex] positive real numbers.
    prove that

    [tex]\lim_{p\to \infty}\left(\frac{x_1^p+...+x_n^p}{n}\right)^{1/p}=max\{x_1,...,x_n\}[/tex]

    can you give me some hints ?
     
  2. jcsd
  3. Nov 8, 2008 #2

    Dick

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    Re: calculus

    Suppose the largest number is x_k. What is the limit of (x_i/x_k)^p? Is that enough of a hint?
     
  4. Nov 8, 2008 #3
    Re: calculus

    this limit equals 1 when x_i=x_k and 0 when x_i<x_k.
    so lim (x_1^p+...+x_n^p)/x_k^p = number of x_i that equal to x_k.
    I don't know how to continue.
     
  5. Nov 8, 2008 #4

    Dick

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    Re: calculus

    Factor x_k^p out of the sum in your limit (that's how you get that expression whose limit you just figured out). Then bring it outside of the ()^(1/p) power as x_k.
     
  6. Nov 8, 2008 #5
    Re: calculus

    Thank you !
     
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