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Large deviation principle and Fibonacci sequence

  1. Aug 25, 2012 #1

    dyh

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    1. The problem statement, all variables and given/known data

    A>0, B can be any number

    2. Relevant equations

    To show that

    lim(n->∞) (1/n)log[A((1+sqrt(5))/2)^n +B((1-sqrt(5))/2)^n] = (1+sqrt(5))/2


    3. The attempt at a solution

    I used Lhopital's Rule to solve this and got
    log((1+sqrt(5))/2)

    So, I don't know what is wrong.
    If you guys know how to prove this one, please let me know.
    Thanks a lot
     
  2. jcsd
  3. Aug 25, 2012 #2

    HallsofIvy

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    Staff Emeritus
    Science Advisor

    If you used L'Hopital's principle, how do you still have a logarithm?
     
  4. Aug 25, 2012 #3

    dyh

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    Because if I differentiate A*(1+sqrt(5))^n with respect to n

    then I would get

    A*log(1+sqrt(5))*(1+sqrt(5))^n
     
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