Continued Fractions problem

  • #1
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1

Homework Statement


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##.


Homework Equations


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


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.
 

Answers and Replies

  • #2
haruspex
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If you set a = 2α and b = β, what are the roots of x2-ax-b?
 
  • #3
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If you set a = 2α and b = β, what are the roots of x2-ax-b?
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!
 
  • #4
arildno
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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!
A deep result in maths is that arithmetic and algebraic results are NOT static over time, but depend on what help you get!
:smile:
 
  • #5
274
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A deep result in maths is that arithmetic and algebraic results are NOT static over time, but depend on what help you get!
:smile:
Amen! Haha.
 

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