1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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

    micromass

    User Avatar
    Staff Emeritus
    Science Advisor
    Education Advisor
    2016 Award

    This thread is 2 years old. Please be more careful before posting in an old thread.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Proving limit of the nth root of n
Loading...