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Show lim((2n)&^(1/n)) = 1

  1. Mar 12, 2012 #1
    1. The problem statement, all variables and given/known data
    I need to show the lim((2n)^(1/n)) = 1


    2. Relevant equations
    I will be using the definition of the limit as well as using the Binomial Theorem as an aide.

    I am following an example from my book quite similar. So applying the Binomial Theorem to this problem, I will choose to write (2n)^(1/n) as 1 + Kn for some Kn > 0.

    Raising both sides to the n power, we have 2n = (1 + Kn)^n ≥ 1 + (1/2)n(n-1)(Kn)^2
    => 2n ≥ (1/2)n(n-1)(Kn)^2
    I will then solve for Kn to get that Kn≤ 2/√n-1

    This tells us that there is some Nε such that 2/√Nε-1 < ε since ε>0 (By the Archimedean Property).

    3. The attempt at a solution
    Now, applying that to my proof, I have:
    Let ε>0 be given.
    0<(2n)^(1/n) -1 = (1 + Kn) -1 = Kn ≤ 2/√n-1 < ε
    Since ε>0 is arbitrary, we can conclude that lim((2n)^(1/n)) = 1

    I appreciate the help! I would like to know if this is mostly correct, and would like help in rewriting it to make it neater. Thank you :)
     
  2. jcsd
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