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Math Help - continued fraction expansion help, / proof

  1. #1
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    continued fraction expansion help, / proof

    I have attached the question I need help, question 5, and really not sure how to go about,

    any help / tips appreciated


    thank you
    Attached Thumbnails Attached Thumbnails continued fraction expansion help, / proof-screen-shot-2013-10-22-11.20.02.png  
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  2. #2
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    Re: continued fraction expansion help, / proof

    Begin with writing out a little bit of the continued fraction:
    y = s + \cfrac{1}{s+ \cfrac{1}{s + \cfrac{1}{s+\cfrac{1}{\ddots}}}}

    Now, you are asked to show that y = s+\dfrac{1}{y}. This is obvious. Just replace everything under the top 1 of the continued fraction by y (since it is equal to y).

    So, how can you solve the rest of it? Well, solve for y. If you multiply everything by y, you get y^2 = sy+1. Treat s as a constant and you have a quadratic of one variable. Solve for y. Then x = 1 + \dfrac{1}{y}. Just plug in whatever you get as your answer.
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  3. #3
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    Re: continued fraction expansion help, / proof

    thank you, but do you mean replace the first top 1 by y = 1 + 1/y?
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  4. #4
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    Re: continued fraction expansion help, / proof

    Quote Originally Posted by Tweety View Post
    thank you, but do you mean replace the first top 1 by y = 1 + 1/y?
    Huh?

    I mean
    y = s + \cfrac{1}{s+ \cfrac{1}{s + \cfrac{1}{s+\cfrac{1}{\ddots}}}}  = s + \cfrac{1}{\left(s+ \cfrac{1}{s + \cfrac{1}{s+\cfrac{1}{\ddots}}}\right)}

    According to the first equality, the part that is in parentheses in the rightmost equation is equal to y.
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