[Math] Find the fraction where the decimal expansion is infinite

Question

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.

Answer 1

Like you hint, we can observe that the value $$0.11235955056\ldots = \sum_{k = 1}^{\infty} 10^{-k} F_k,$$ where $F_k$ is the $k$th Fibonacci number (with the convention that $F_0 = 0$, $F_1 = 1$), is simply the value of the series $$F(x) := \sum_{k = 1}^{\infty} F_k x^k = x + x^2 + 2x^3 + 3x^4 + 5x^5 + 8x^6 + 13x^7 + \cdots$$ at $x = \frac{1}{10}$.

Hint Using the defining recurrence relation $$F_{k + 2} = F_{k + 1} + F_k$$ (and the above convention) gives that $F(x)$ satisfies $$F(x) = x + x F(x) + x^2 F(x).$$

Rearranging gives that on the open interval of convergence of the series, which turns out to be $(-1/\phi, 1/\phi)$---where $\phi$ is the Golden Ratio, and which in particular contains the value $\frac{1}{10}$ of interest)---we have $$F(x) = \frac{x}{1 - x - x^2}.$$ Thus, the series has value $$F\left(\tfrac{1}{10}\right) = \frac{\left(\frac{1}{10}\right)}{1 - \left(\frac{1}{10}\right) - \left(\frac{1}{10}\right)^2} = \color{#bf0000}{\boxed{\frac{10}{89}}} .$$ Since this is a rational number, its decimal expansion repeats: $$0.\overline{1123595505617977528089887640449438202247191}.$$