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Math Help - Fibonacci Power Series

  1. #1
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    Fibonacci Power Series

    I am having trouble with this question:
    Define F(x)=$\displaystyle\sum\limits_{n=0}^\infty F_nx^n$ where F_n is the nth Fibonacci number. Determine the radius of convergence of F(x). Using F_{n+2} = F_{n+1} + F_n find F(x) in closed form.

    I think I have managed to do the radius of convergence but I don't know if I am correct. Is the answer the solution to x^2-x-1 i.e. \frac{2}{1+\sqrt{5}}?

    As for the second part I have completely no idea and would appreciate some help starting the question (I should note I have tried writing F(x) out in terms of the formula but have not gotten anywhere with this)
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  2. #2
    MHF Contributor chisigma's Avatar
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    Quote Originally Posted by worc3247 View Post
    I am having trouble with this question:
    Define F(x)=$\displaystyle\sum\limits_{n=0}^\infty F_nx^n$ where F_n is the nth Fibonacci number. Determine the radius of convergence of F(x). Using F_{n+2} = F_{n+1} + F_n find F(x) in closed form.

    I think I have managed to do the radius of convergence but I don't know if I am correct. Is the answer the solution to x^2-x-1 i.e. \frac{2}{1+\sqrt{5}}?

    As for the second part I have completely no idea and would appreciate some help starting the question (I should note I have tried writing F(x) out in terms of the formula but have not gotten anywhere with this)
    The explicit form for the Fibonacci's number is...

    \displaystyle F_{n}= \frac{\varphi^{n} - (-1)^{n}\ \varphi^{-n}}{\sqrt{5}} (1)

    ... where...

    \displaystyle \varphi= \frac{1+\sqrt{5}}{2} (2)

    ... so that is...

    \displaystyle F(x)= \sum_{n=0}^{\infty} F_{n}\ x^{n} = \frac{1}{\sqrt{5}}\ \{\sum_{n=0}^{\infty} (\varphi\ x)^{n} - \sum_{n=0}^{\infty} (-\frac{x}{\varphi})^{n}\} (3)

    Now (3) is the sum of two series... the first converges for |x|<\varphi^{-1} , the second for |x|<\varphi and is \varphi^{-1}< \varphi so that...

    Kind regards

    \chi \sigma
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  3. #3
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    Do you then just take geometric series and get:
    F(x)=\frac{1}{\sqrt{5}}(\frac{1}{1-\varphi x} - \frac{\varphi}{\varphi + x})?
    Cheers
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  4. #4
    MHF Contributor chisigma's Avatar
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    Quote Originally Posted by worc3247 View Post
    Do you then just take geometric series and get:
    F(x)=\frac{1}{\sqrt{5}}(\frac{1}{1-\varphi x} - \frac{\varphi}{\varphi + x})?
    Cheers
    Yes!... it does!...

    Kind regards

    \chi \sigma
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