Let's just say you've been given an assignment to write a number like 0.435435435... in expanded notation. How would you proceed? And how to generalize something like this with algebra?
Thank you very much.
EDIT
Example of expanded notation of a number is on the right side of the following equation:
$0.435 = \left(4\times{}10^{-1}\right) + \left(3\times{}10^{-2}\right) + \left(5\times{}10^{-3}\right)$
The problem is that as soon as we draw a line above the entire sum of these products to try to denote repetition in algebraic form, the line would be perceived (unlike intended) as a grouping symbol.
I apologize for not clarifying things initially.
Thank you.
Let $x = \smash[t]{0.\overline{435}}$. Since the pattern repeats every three digits, we can express the number as a geometric sum and find its exact (rational) value that way. Observe: \begin{align} &0.\color{red}{435}\color{green}{435}\color{blue}{435}\dots \\ \hline &0.\color{red}{435} \\ {}+{}\; &0.000\color{green}{435} \\ {}+{}\; &0.000000\color{blue}{435} \\ {}+{}\; &\phantom{0.000000000}\cdots \end{align}
Using the ancient geometric sum formula (where $|r| < 1$),
$$ a + ar + ar^2 + ar^3 + \cdots = \frac{a}{1-r}, $$ we calculate \begin{align} x &= \frac{435}{1000^{\phantom{1}}} + \frac{435}{1000^2} + \frac{435}{1000^3} + \cdots \\[4pt] &= \frac{\frac{435}{1000}}{1 - \frac{1}{1000}} \\[4pt] &= \frac{435}{999} \end{align} It happens that this fraction reduces, so we might want to write $$ x = \frac{145}{333}. $$
This generalizes as you might expect, where the denominator ends up being a string of $n$ consecutive $9$s, i.e. the integer $10^n - 1$, where $n$ is the length of the repeating block.
So, for instance, with a block of length $n=2$, the denominator is $10^2-1 = 99$: $$ 0.\overline{47} = 0.474747\dots = \frac{47}{99} $$