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: Limit and error term question

  1. Feb 21, 2007 #1
    1. The problem statement, all variables and given/known data

    Compute the limit a_n=n^(1/n) without using the fact that lim log a_n=log lim a_n. Instead, we're expected to solve this using binomial theorem and the error term.


    2. Relevant equations

    na

    3. The attempt at a solution

    Well, the error term = |a_n - L| which is expected to go to zero. I tried using binomial theorem on ((n-1)+1)^1/n with no luck. Also tried squeeze theorem.
     
  2. jcsd
  3. Feb 23, 2007 #2
    hint

    The binomial theorem is not obvious since you do now have whole number exponents.

    Here is an attempt:
    a_n=n^{1/n}

    a_{n+1}-a_n = (n+1)^{1/(n+1)}-n^{1/n}

    <(n+1)^{1/n}-n^{1/n}


    and a_{n+1}-a_n > (n+1)^{1/(n+1)}-(n)^{1/(n+1)}
    Showing that these comparitive terms go to 0 as n goes to infinity may use the "Binomial Number" expansion. Then, Sandwich Thm applies.

    http://en.wikipedia.org/wiki/Binomial_theorem
     
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook