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Math Help - Continued fraction

  1. #1
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    Continued fraction

    i need help to solve this exercise
    To prove the continued fraction of (d^2 +1)^(1/2) is [d;2d,2d,2d,….]
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  2. #2
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    Hello, user!

    Prove that the continued fraction of . \sqrt{d^2 +1} . is . \bigg[d,2d,2d,2d, \hdots\bigg]

    \text{Let: }\;x \;=\;d + \frac{1}{2d + \dfrac{1}{2d + \hdots}}

    \text{Add }d\text{ to both sides: }\;x + d \;=\;2d + \frac{1}{\left\{2d + \dfrac{1}{2d + \hdots}\right\}} \;\begin{array}{c} \\ \\   \Leftarrow\text{ This is }(x+d) \end{array}

    \text{Then we have: }\;x + d \;=\;2d + \frac{1}{x+d}


    \text{Multiply by }(x+d)\!:\;\;(x+d)^2 \;=\;2d(x+d) + 1 \quad\Rightarrow\quad x^2 + 2dx + d^2 \;=\;2dx + 2d^2 + 1


    \text{Therefore: }\;x^2 \;=\;d^2+1 \quad\Rightarrow\quad x \;=\;\sqrt{d^2+1}

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  3. #3
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    answer

    Hi friend. Thanks for your answer.
    I had another solution finding
    alpha zero, a0 to obtain that a 3 this is the integer part of alpha 3 is [d; 2d,2d,2d,... ] but this solution is a little more larger than your solution
    Bye see you soon
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  4. #4
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    Hi friends. How can i prove that the neutral element to the sum is unique.
    Thank you
    Sincrely, user
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