1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Continued Fractions problem

  1. Sep 18, 2013 #1
    1. The problem statement, all variables and given/known data
    Let x be any positive real number and suppose that ##x^2-ax-b=0## where ##a,b## are positive. I would like to use the equation that I provided in relevant equations which I proved to prove that
    $$
    \sqrt{\alpha^{2}+\beta}=\alpha+\cfrac{\beta}{2 \alpha+\cfrac{\beta}{2 \alpha+\cfrac{\beta}{2\alpha+\ddots}}}
    $$
    where ##\alpha,\beta>0##.


    2. Relevant equations
    I proved that
    $$
    x=a+\cfrac{b}{a+\cfrac{b}{a+\cfrac{b}{a+\ddots}}}.
    $$


    3. The attempt at a solution
    I tried to do things like find values of ##a,b## so that when I transformed the equation ##x^{2}-ax-b=0## into a continued fraction that I would get the desired continued fraction with ##x=\sqrt{\alpha^{2}+\beta}## but that didn't work out.

    I also tried changing ##\sqrt{\alpha^{2}+\beta}## directly into a continued fraction using the canonical continued fraction algorithm but I then had to consider different values of ##\beta## which would give me different continued fractions that I didn't really know how to combine to create the desired continued fraction.

    I tried to plug ##x=\sqrt{\alpha^{2}+\beta}## into ##x^{2}-ax-b=0## and then solve for ##a,b## but that didn't get too far with two variables and one equation.
     
  2. jcsd
  3. Sep 19, 2013 #2

    haruspex

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member
    2016 Award

    If you set a = 2α and b = β, what are the roots of x2-ax-b?
     
  4. Sep 19, 2013 #3
    I don't know why but I remember trying that and it didn't work but now that I try it again after you mention it, it works. Thanks!
     
  5. Sep 19, 2013 #4

    arildno

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member
    Dearly Missed

    A deep result in maths is that arithmetic and algebraic results are NOT static over time, but depend on what help you get!
    :smile:
     
  6. Sep 19, 2013 #5
    Amen! Haha.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted



Similar Discussions: Continued Fractions problem
  1. Continued Fractions. (Replies: 1)

  2. Continued Fractions (Replies: 4)

Loading...