Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: Proving limit of the nth root of n

  1. Sep 4, 2010 #1
    1. The problem statement, all variables and given/known data

    Prove the following limit:

    [tex]lim_{n \rightarrow \infty } n^{ 1 / n } = 1 [/tex]

    2. Relevant equations

    Not sure.

    3. The attempt at a solution

    Given any [tex]\epsilon > 0[/tex], choose [tex]N \in \mdseries N[/tex] s.t.

    [tex]\left| n^{ 1 / n } - 1 \right| < \epsilon[/tex] for all [tex]n > N[/tex]

    I am not sure how to proceed.
     
  2. jcsd
  3. Sep 4, 2010 #2

    jgens

    User Avatar
    Gold Member

    If you're still working on this problem and need to do it with epsilons and deltas, I think that choosing N = exp(log(1+ε)-1) should suffice. I can't find a nice/elegant epsilon delta solution to this problem, but maybe someone else can.
     
    Last edited by a moderator: Sep 5, 2010
  4. Jul 11, 2012 #3
    Prove [tex]\lim _{n\to \infty} \sqrt[n]{n}=1[/tex].
    Proof: We want:
    [tex]
    |\sqrt[n]{n}-1|<\epsilon
    [/tex]
    The abs sign can be safely dropped, it follows that
    [tex]
    n<(1+\epsilon)^n
    [/tex]
    Using binomial theorem to expand the first 3 terms of RHS.
    [tex]
    n<1+n\epsilon+\frac{1}{2}n(n-1)\epsilon^2+...
    [/tex]
    As long as we make n<0.5n(n-1)εε, the first inequality holds. It requires
    [tex]
    n>1+\frac{2}{\epsilon^2}
    [/tex]

    With all that said,
    For any ε>0, there exists N=[1+2/(εε)], such that if n>N, then
    [tex]
    |\sqrt[n]{n}-1|<\epsilon
    [/tex]

    Q.E.D
    P.S. I love ε-δ proof:)
     
    Last edited: Jul 11, 2012
  5. Jul 11, 2012 #4
    This thread is 2 years old. Please be more careful before posting in an old thread.
     
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook