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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


    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

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

    Here is an attempt:

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

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