How do I show convergence of continued fraction?

Question

How can I show convergence of the following continued fraction?

$3+\cfrac{3}{3+\cfrac{3}{3+\cfrac{3}{3+\cfrac{3}{\cdots}}}}$

Thank you in advance.

Answer 1

Lets take the sequence $x_n=3+\frac{3}{x_{n-1}}$ and $x_1=3$ In case of convergence the limit of the sequence will verifie the following equation $$L=3+\frac{3}{L}\implies L^2=3L+3\iff L=\frac{3\pm\sqrt{21}}{2}$$ And it must be $L=\frac{3+\sqrt{21}}{2}$, because the sequence is strictly positive. To prove the convergence we need monotony and boundedness. The monotony is easy to prove, is strictly increasing. Now we are gonna see the boundedness by some bound like for example $4$.

$x_1=3<4$ and $x_{k+1}<4\iff 3+3/x_k<4\iff 3<x_{k}$. That is true because $x_{n}$ is strictly increasing and $x_1=3$ So $x_n<4$,$\forall n\in\mathbb{N}$. Thus is convergent and $$L=3+\frac{3}{3+\frac{3}{3+\frac{3}{3+\frac{3}{...}}}}=\frac{3+\sqrt{21}}{2}$$