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Convergence of n^(1/n)

  1. Oct 2, 2008 #1


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    1. The problem statement, all variables and given/known data
    Let [tex]x_n=n^{\frac{1}{n}}[/tex] be a sequence. Does it converge, and if so to what value?

    2. Relevant equations

    3. The attempt at a solution
    Clearly it will converge to 1, so that's what we want to prove.
    [tex]\forall \epsilon>0\exists N \textnormal{s.t.} m>N \rightarrow |m^{\frac{1}{m}}-1| < \epsilon.[/tex]

    ...And then I am stuck... any hints on how to select N?
  2. jcsd
  3. Oct 3, 2008 #2
    You could try and find a weaker case for selecting N. For example, If [tex]\epsilon[/tex], N>0

    [tex]N < (\epsilon + 1)^{N-1} < (\epsilon + 1)^{N}[/tex]. Can you see where I'm going with this?
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