How to Compute Infinite Continued Factions

Question

I'm supposed to find the value of the infinite continued fracton $[2;1,3,1,3,1,3,1,3...]$. How would I go about doing this?

Answer 1

Denote the continued fraction as $x$ and $x-2=y$, i.e.

$$y=\cfrac1{1+\cfrac1{3+\cdots}}$$

Moreover we get that

\begin{align*} y=\cfrac{1}{1+\cfrac{1}{3+y}}\implies y&=\frac1{\cfrac{4+y}{3+y}}\\ &=\frac{3+y}{4+y} \end{align*}

Which overall leads to the quadratic equation

$$y^2+3y-3=0$$

Solving this equation and choosing the positive solution further leads to

$$y=\frac{-3+\sqrt{21}}2\implies x=\frac{-3+\sqrt{21}}2+2$$

$$\therefore~x~=~2+\cfrac1{1+\cfrac1{3+\cdots}}~=~\frac{1+\sqrt{21}}2$$

The solution is confirmed by this calculator which produces the given continued fraction for the resulting value of $x$.