Ones, Twos And Threes

Let x = 0.123 123 123 ....................... ---- 1
Then 1000x = 123.123 123 123 ........................... ------- 2
Subtacting (1) from (2) , we get :
1000 x - x = (123.123 123 123 ...................) - (0.123 123 123 ............................)
or 999x = 123
or x = $\frac{123}{999}$ = $\frac{41}{333}$ Q.E.D

Relevant wiki: Converting Decimals and Fractions

$\underline{\textbf{Lemma:}}$ $0.\overline{a_1a_2a_3\cdots a_n}=\frac{\overline{a_1a_2a_3\cdots a_n}}{\underbrace{99\cdots 999}_{n\;9's}}$

$\text{Using the lemma with}\; n=3\;\text{and}\;\overline{a_1a_2a_3}=123,\text{we get that:}\\ 0.123123123\cdots=\frac{123}{999}=\color{#20A900}{\boxed{\boxed{\frac{41}{333}}}}$

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$\underline{\textbf{Proof of Lemma:}}$

$\begin{aligned}\text{Let}\quad x&=0.\overline{a_1a_2a_3\cdots a_n}\\ \implies 10^{n}x&=\overline{a_1a_2a_3\cdots a_n}.\overline{a_1a_2a_3\cdots a_n}\\ 10^{n}x-x&=\overline{a_1a_2a_3\cdots a_n}.\overline{a_1a_2a_3\cdots a_n}-0.\overline{a_1a_2a_3\cdots a_n}\\ (10^{n}-1)x&=\overline{a_1a_2a_3\cdots a_n}\\ x&=\boxed{\frac{\overline{a_1a_2a_3\cdots a_n}}{10^{n}-1}}\end{aligned}$

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$\underline{\textbf{Notations:}}$

$\overline{X_1X_2X_3\cdots X_n}\;\text{denotes the number formed by the concatenation of the digits} \; X_1,X_2,X_3,\cdots,X_n$

The number $0.123123123123...= 0.123 + 0.000123 + 0.000000123 + ...$

With that in mind, we can write the number $0.123123123123...$ as a geometric series. Therefore, we can find our fractional equivalent by using the formula:

$S_n$ = $a_1$ $\frac{1-r^n}{1-r}$

$S_n$ = Sum $a_1$ = First Term $r$ = Common Ratio $n$ = Number of Elements

Since we are looking for an approximation, let's suppose we use 100 of those summands in the infinite sum above. For our solution, the values for the variables are as follows:

$S_n$ = 0.123123123123 $a_1$ = 0.123 $r$ = 1/1000 $n$ = 100

Plug them into the formula

$0.123123123123$ = $0.123$ $\frac{1-(1/1000)^{100}}{1-1/1000}$

In the numerator, we will be left with 1, since subtracting 0.0000000000000000000... from 1 will give you 1. (See this link for more info)

$0.123$ ( $\frac{1}{1000/1000-1/1000}$ ) = $0.123$ ( $\frac{1}{999/1000}$ )

= $\frac{0.123}{999/1000}$ = $\frac{123}{999}$

$\boxed{41/333}$