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Math Help - Fibonacci decimal

  1. #1
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    Fibonacci decimal

    Find the value of this infinite decimal:
    Code:
          0.01
          0.001
          0.0002
          0.00003
          0.000005
          0.0000008
          0.00000013
          0.000000021
        +     ...
        -----------------
    S  =  0.011235955 ...
    That is: . S \;=\;\sum^{\infty}_{n=1} \frac{F_n}{10^{n+1}}

    . . where F_n is the n^{th} Fibonacci number.

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  2. #2
    Forum Admin topsquark's Avatar
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    Quote Originally Posted by Soroban View Post
    Find the value of this infinite decimal:
    Code:
          0.01
          0.001
          0.0002
          0.00003
          0.000005
          0.0000008
          0.00000013
          0.000000021
        +     ...
        -----------------
    S  =  0.011235955 ...
    That is: . S \;=\;\sum^{\infty}_{n=1} \frac{F_n}{10^{n+1}}

    . . where F_n is the n^{th} Fibonacci number.

    Well,
    F_n = \frac{1}{\sqrt{5}} \left ( \frac{1 + \sqrt{5}}{2} \right ) ^n - \frac{1}{\sqrt{5}} \left ( \frac{1 - \sqrt{5}}{2} \right ) ^n

    So S is the difference of two geometric series:
    S = \frac{1}{10\sqrt{5}}\sum_{n = 1}^{\infty} \left ( \frac{1 + \sqrt{5}}{2 \cdot 10} \right ) ^n - \frac{1}{10\sqrt{5}}\sum_{n = 1}^{\infty} \left ( \frac{1 - \sqrt{5}}{2 \cdot 10} \right ) ^n

    I have to get to dinner, so I'll let someone else finish. It's trivial from here anyway.

    -Dan
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  3. #3
    Eater of Worlds
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    Hey Soroban. I think I may have this one. Let's see. I encountered this proof as a homework assignment back in college. In a Seminars in Mathematics course.

    [tex]\frac{1}{10}\sum_{n=1}^{\infty}\frac{F_{n}}{10^{n} }[\math]

    Let d=0.11235955056........=\frac{F_{1}}{10}+\frac{F_{  2}}{10^{2}}+\frac{F_{3}}{10^{3}}+.........

    Now use 10d+d and we have:

    11d=F_{1}+\frac{F_{2}+F_{1}}{10}+\frac{F_{3}+F_{2}  }{10}+\frac{F_{4}+F_{3}}{10}+....

    =1+\frac{F_{3}}{10}+\frac{F_{4}}{10^{2}}+\frac{F_{  5}}{10^{3}}+..........

    =1+100d-10F_{1}-F_{2}

    =1+100d-10-1

    =100d-10

    So, we have:

    11d=100d-10

    89d=10

    d=\frac{10}{89}

    Don't forget to multiply by our 1/10 and get:

    \frac{1}{89}
    Last edited by galactus; June 25th 2007 at 08:58 AM.
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  4. #4
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    Welcome back, you were away for a long time!
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  5. #5
    Eater of Worlds
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    Thanks PH. Yeah, it has been a while. I see you're up to 6200. Very prolific

    How's the academic world treating you?.
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  6. #6
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    Quote Originally Posted by galactus View Post
    How's the academic world treating you?.
    I am on vacation now. 2 Months to go.
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  7. #7
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    Exclamation

    Quote


    /Quote

    I'm pretty sure that F(4)/10 should be F(4)/100, etc.
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  8. #8
    Eater of Worlds
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    You're so right, G. Thanks for the catch. I fix.
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  9. #9
    Forum Admin topsquark's Avatar
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    Quote Originally Posted by topsquark View Post
    Well,
    F_n = \frac{1}{\sqrt{5}} \left ( \frac{1 + \sqrt{5}}{2} \right ) ^n - \frac{1}{\sqrt{5}} \left ( \frac{1 - \sqrt{5}}{2} \right ) ^n

    So S is the difference of two geometric series:
    S = \frac{1}{10\sqrt{5}}\sum_{n = 1}^{\infty} \left ( \frac{1 + \sqrt{5}}{2 \cdot 10} \right ) ^n - \frac{1}{10\sqrt{5}}\sum_{n = 1}^{\infty} \left ( \frac{1 - \sqrt{5}}{2 \cdot 10} \right ) ^n

    I have to get to dinner, so I'll let someone else finish. It's trivial from here anyway.

    -Dan
    Okay, I'll finish.

    We have
    \sum_{n = 0}^{\infty}ar^n = a + \sum_{n = 1}^{\infty}ar^n = \frac{a}{1 - r}

    So
    \sum_{n = 1}^{\infty}ar^n = <br />
\frac{a}{1 - r} - a

    Thus
    \sum_{n = 1}^{\infty} \left ( \frac{1 + \sqrt{5}}{2 \cdot 10} \right ) ^n = \frac{1}{1 - \left ( \frac{1 + \sqrt{5}}{2 \cdot 10} \right )} - 1

    and
    \sum_{n = 1}^{\infty} \left ( \frac{1 - \sqrt{5}}{2 \cdot 10} \right ) ^n = \frac{1}{1 - \left ( \frac{1 - \sqrt{5}}{2 \cdot 10} \right )} - 1

    So
    S = \frac{1}{10\sqrt{5}}\sum_{n = 1}^{\infty} \left ( \frac{1 + \sqrt{5}}{2 \cdot 10} \right ) ^n - \frac{1}{10\sqrt{5}}\sum_{n = 1}^{\infty} \left ( \frac{1 - \sqrt{5}}{2 \cdot 10} \right ) ^n = \frac{1}{10\sqrt{5}} \left ( \frac{1}{1 - \left ( \frac{1 + \sqrt{5}}{2 \cdot 10} \right )} - \frac{1}{1 - \left ( \frac{1 - \sqrt{5}}{2 \cdot 10} \right )} \right )

    S = \frac{1}{10\sqrt{5}} \left ( \frac{20}{19 - \sqrt{5}} - \frac{20}{19 + \sqrt{5}} \right )

    S = \frac{2}{\sqrt{5}} \left ( \frac{19 + \sqrt{5}}{19^2 - 5} - \frac{19 - \sqrt{5}}{19^2 - 5} \right )

    S = \frac{2}{\sqrt{5}} \left ( \frac{2 \sqrt{5}}{356} \right )

    S = \frac{4}{356} = \frac{1}{89}

    as galactus already informed us.

    -Dan
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  10. #10
    Eater of Worlds
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    Cool, Topsquark. That's a good one.

    One could also just use the closed-form for the Fibonacci numbers with x=1/10.

    \frac{x}{1-x-x^{2}}\Rightarrow\frac{\frac{1}{10}}{1-(\frac{1}{10})-(\frac{1}{10})^{2}}=\frac{10}{89}

    Multiply by our 1/10 and we get 1/89.
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