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Proving an inequality involving exponentiation

  1. Jul 10, 2010 #1
    1. The problem statement, all variables and given/known data

    Show that [tex]\left( 1 - \frac{\ln n}{kn} \right)^n > \frac{1}{n^{1/k} + 1}[/tex] holds for all integers [tex]n\geq 1[/tex] and [tex]k\geq 2[/tex].

    3. The attempt at a solution

    I first tried to find a proof for [tex]k=2[/tex] by showing that the quotient LHS/RHS goes to 1 and has negative slope everywhere, but this becomes rather unwieldy.
  2. jcsd
  3. Jul 10, 2010 #2


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    Homework Helper

    I would be tempted to try a proof by induction on n. Show that the inequality holds for n=1, Assume there is an integer m for which the inequality holds and show it's true for m+1.

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