TITLE: Find the fraction where the decimal expansion is infinite?
QUESTION [6 upvotes]: Find the fraction with integers for the numerator and denominator, where the decimal expansion is $0.11235.....$ 
The numerator and denominator must be less than $100$. 
Find the fraction. 
I believe I can use generating functions here to get $1+x+2x^2+3x^3+5x^4+.....$, but I do not know how to apply it.

REPLY [6 votes]: Continued fractions:
\begin{align}
\frac{11235}{100000}&=\frac{2247}{20000}\\
&=\cfrac{1}{8+\cfrac{2024}{2247}}\\
&=\cfrac{1}{8+\cfrac{1}{1+\cfrac{223}{2024}}}\\
&=\cfrac{1}{8+\cfrac{1}{1+\cfrac{1}{9+\cfrac{1}{13+\cfrac{2}{17}}}}}\\
&=\cfrac{1}{8+\cfrac{1}{1+\cfrac{1}{9+\cfrac{1}{13+\cfrac{1}{8+\cfrac{1}{2}}}}}}\\
&=[0;8,1,9,13,2]
\end{align}
The first convergent is $1/8$; the second convergent is
$$
\cfrac{1}{8+1}=\frac{1}{9};
$$
the third convergent is
$$
\cfrac{1}{8+\cfrac{1}{1+\cfrac{1}{9}}}=\cfrac{1}{8+\cfrac{9}{10}}=\frac{10}{89};
$$
The fourth convergent can be computed to $131/1166$ (see this continued fraction calculator)