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Radius of Convergence of Fibonacci sequence

  1. May 13, 2010 #1
    Radius of Convergence of Fibonacci sequence :)

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

    Given the Fibonacci sequence where

    [tex]\frac{1}{1-x-x^2} = \sum_{n=0}^{\infty} F_{n} x^n[/tex]

    find the radius of convergence around zero.

    2. Relevant equations

    Ratio test

    3. The attempt at a solution

    By the radio test [tex] \lim_{n \to \infty} \left|\frac{F_{n+1}x^{n+1}}{F_{n}x^n}\right| =|x| = 0[/tex] Thus the radius of Convergence since R = 0.

    Am I correct?
     
    Last edited: May 13, 2010
  2. jcsd
  3. May 13, 2010 #2

    jbunniii

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    Re: Radius of Convergence of Fibonacci sequence :)

    How did you get that? It's not true.

    [tex]\frac{F_{n+1}}{F_n}[/tex]

    converges to a certain positive number (the so-called "golden section") as [itex]n \rightarrow \infty[/itex]. What does that say about

    [tex]\lim_{n \rightarrow \infty} \left|\frac{F_{n+1}x^{n+1}}{F_n x^n}\right|?[/tex]
     
  4. May 13, 2010 #3
    Re: Radius of Convergence of Fibonacci sequence :)

    I will look at it again, sorry :)
     
  5. May 13, 2010 #4
    Re: Radius of Convergence of Fibonacci sequence :)

    You can also find the zeros of the generating function on the left. The one with the smalles absolute value (smallest distance from the origin around which you perform your Taylor expansion) should give you the radius of convergence.
     
  6. May 14, 2010 #5
    Re: Radius of Convergence of Fibonacci sequence :)

    I can see I was way of and I'm sorry for that. Golden ratio and all that. I didn't learn about that mathwise in High School here in my part of the world. So I will read up on it :)

    However I'm now told that the Fibonacci sequence convergeces towards

    the socalled golden ratio [tex]\frac{1+\sqrt{5}}{2}[/tex]
     
  7. May 14, 2010 #6

    HallsofIvy

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    Re: Radius of Convergence of Fibonacci sequence :)

    Well, not the Fibonacci sequence itself- that increases without bound. It is the ratio
    [tex]\frac{F_{n+1}}{F_n}[/tex]
    that converges to the "golden ratio"
    [tex]\frac{1+ \sqrt{5}}{2}[/tex]
     
  8. May 14, 2010 #7
    Re: Radius of Convergence of Fibonacci sequence :)

    But your radius of convergence is not that.
     
  9. May 14, 2010 #8
    Re: Radius of Convergence of Fibonacci sequence :)

    HallsofIvy we all know you are like a Jedi then it comes to Math and Science compared to rest of us.

    So to show the radius of convergence of the fibunacci sequence do I need to show first that whole thing convergences to the golden ratio?
     
  10. May 16, 2010 #9
    Re: Radius of Convergence of Fibonacci sequence :)

    so for the ratio I get

    that [tex]\lim_{n \to \infty} |\frac{F_{n+1}}{F_n}| = \frac{1+\sqrt{5}}{2}[/tex]

    thus the radius of convergence [tex]R = \frac{1+\sqrt{5}}{2}[/tex]

    how is that?
     
  11. May 16, 2010 #10
    Re: Radius of Convergence of Fibonacci sequence :)

    It is not correct. The general element of the series is not [itex]F_{n}[/itex], but [itex]F_{n} x^{n}[/itex].
     
  12. May 16, 2010 #11
    Re: Radius of Convergence of Fibonacci sequence :)

    this maybe be a stupid question but if as a previous post surgests I do Taylor series expansion

    F_0 + F_1 z + F_2 z^2 + .........+ F_(n+1) z^(n+1)

    isn't the ratio which I need to find

    lim = (F_(n+1) z^(n+1)/F_nx^n) = ???

    for n - > infty
     
  13. May 16, 2010 #12
    Re: Radius of Convergence of Fibonacci sequence :)

    You need to find the constraints on [itex]z[/itex], and you have not done so.
     
  14. May 16, 2010 #13
    Re: Radius of Convergence of Fibonacci sequence :)

    Its suppose to be the radius of convergence around zero but if I set z to zero then the whole fraction turns zero. That can't the right can't it?
     
  15. May 17, 2010 #14
    Re: Radius of Convergence of Fibonacci sequence :)

    Please use the ratio test correctly as you did in your original post, although you made a mistake evaluating the relevant limit there.

    EDIT:

    On further inspection, I don't know why you equate the limit to zero?
     
  16. May 17, 2010 #15
    Re: Radius of Convergence of Fibonacci sequence :)

    so my original post was right?
     
  17. May 17, 2010 #16
    Re: Radius of Convergence of Fibonacci sequence :)

    No. Please refer to the proper use of the ratio test for any power series.
     
  18. May 17, 2010 #17
    Re: Radius of Convergence of Fibonacci sequence :)

    we have the power series:

    [tex]\sum_{n=1} a_{n} x^{n-1}[/tex]

    from 1 to infinity, where [tex] a_{n} [/tex] is the nth term of the fibonacci sequence.

    From the ratio test, we have the form

    [tex] a_{n+1} / a_{n} * x [/tex] and we need this to be < 1 to determine our radius of convergence.. So, we must decide how [tex] a_{n+1} / a_{n} [/tex] behaves.

    We have a0 = 1, a1 = 1 , a2 = 2, a3 = 3

    a1 / a0 = 1 , a2 / a1 = 2, a3 / a2 = 3/2 .. do this enough and convince yourself that the ratio between two consecutive an+1 / an is bounded above by something. Prove by induction.
     
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