Computing periodic continued fractions.

Question

Compute $[1,2,3,\overline{1,4}]$ where $\overline{1,4}$ is the periodic part. I looked into explanations about that, but haven't come by an actual algorithm of computing such a thing. I know it is somehow: $$1+{1\over {2+{1\over 3+{1\over 1+{1\over 4+{1\over 1+{1\over 4}..............}}}}}}$$ but I can't see how I turn it into a numbrt. I could really use your help on this.

Answer 1

Note that the period can be written as: $$ x = 1+\frac{1}{4+\frac{1}{x}} $$ At this point solve the quadratic: $$ 4x^2-4x-1 = 0 \Rightarrow x = \frac{1}{2} \pm \frac{\sqrt{2}}{2} $$ and plug the positive solution into: $$ 1 + \frac{1}{2+\frac{1}{3+\frac{1}{x}}} $$