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

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.

Share this great discussion with others via Reddit, Google+, Twitter, or Facebook