-1

I'm interested in solving the equation $$ \color{red}{x}=1+\cfrac{a_1}{b_1+\cfrac{a_2}{b_2+\ddots \cfrac{a_n}{b_n+\color{red}{x}}}}, $$ where $a_i,b_i$ are positive real numbers. Is there a formula to simplify this continued fraction?

Any help will be appreciated.

metamorphy
  • 31,285
  • 15
  • 38
  • 97
Wang
  • 314
  • 3
  • 10
  • Please show the own effort to solve the issue, it is the way it works. In case $a_1=a_2=\dots=a_n=1$ we have indeed something called continued fraction, $$x = [1;\overline{b_1,b_2,\dots,b_{n-1}, b_n+1}]\ ,$$or something like this. At any rate, inductively simplifying the R.H.S. or writing it as a composition of corresponding Möbius / homographic transformations, we obtain an equation of degree two satisfied by $x$. In this generality, no more specific information can be given. Which is in fact the application for the above? Where did this problem occur? – dan_fulea Jan 17 '21 at 17:44

1 Answers1

0

$$b_{n-1}+\frac{a_n}{b_n+x}=\frac{b_{n-1}b_n+a_n+b_{n-1}x}{b_n+x}.$$

From this you can conclude that your equation has the form

$$x=\frac{px+q}{rx+s}$$ where the coefficients can be computed by recurrence. This equation can be rewritten as quadratic.