1. Limited time only! Sign up for a free 30min personal 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!

Convergence n^(1/n)

  1. Jun 1, 2010 #1
    Hello, I was wondering how to proof [tex]a_n = n^{1/n} \to 1.[/tex]

    Doing it straight from the definition got me nowhere. But I was thinking. It is obvious that [tex]\liminf a_n \geq 1[/tex] (since otherwise for big n you could get n^(1/n) < 1 <=> n < 1). And I also have already a proof of [tex]\limsup x_n^{1/n} \leq \limsup \frac{x_{n+1}}{x_n}[/tex] (which is a general result for any row x_n).

    But proving [tex]\limsup \frac{a_{n+1}}{a_n} \leq 1[/tex] seemed to be harder than I thought.

    So I'm completely stuck. Any ideas?

    Thank you,
    mr. vodka
  2. jcsd
  3. Jun 1, 2010 #2
    I don't think you need to use lim sup/inf here.


    So you just need to show that ln(n)/n-->0. Try using l'Hopital's rule.
  4. Jun 1, 2010 #3
    Hm, I'd like to proof it without the use of the exponential function. It's namely introduced in my Analysis course before the exp function, and it's actually used in a proof about power series, which is later used to introduce the e-function. Thanks for your help!
  5. Jun 1, 2010 #4


    User Avatar
    Science Advisor

    If you don't want to use exp function, how about using ln (natural log)?
    ln(an) = ln(n)/n -> 0.
  6. Jun 1, 2010 #5
    Well, we defined that as the inverse of e. Any possibilities without e (or ln)? I appreciate the help though! I find the ln/e proofs very elegant, but I hope you understand I'm going to choose for logical consistency in my course :)
  7. Jun 1, 2010 #6


    Staff: Mentor

    I'm not sure that there's a way around that doesn't use exp or ln.
  8. Jun 1, 2010 #7
    Oh... But it was left as an exercise for us in our course, so there must be. Hmmm, maybe I should contact one of the math assistants for this one then. Thank you guys for your time.
  9. Jun 2, 2010 #8
    For those that are interested: http://myyn.org/m/article/limit-of-nth-root-of-n/ [Broken]
    Last edited by a moderator: May 4, 2017
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook