To express the infinite decimal expansion \( 0.111111\ldots 0.111111\ldots 0.111111\ldots \) as a fraction, we start by recognizing that this expression can be interpreted as \( 0.111111\ldots \), which is a repeating decimal.
The repeating decimal \( 0.111111\ldots \) can be represented as follows:
Let \( x = 0.111111\ldots \).
To eliminate the repeating part, we can multiply \( x \) by 10:
\[ 10x = 1.111111\ldots \]
Now subtract the original equation from this new equation:
\[ 10x - x = 1.111111\ldots - 0.111111\ldots \]
This simplifies to:
\[ 9x = 1 \]
Now, solve for \( x \):
\[ x = \frac{1}{9} \]
Since the entire expression consists of three repetitions of \( 0.111111\ldots \) concatenated together, we can directly write it as:
\[ 0.111111\ldots 0.111111\ldots 0.111111\ldots = \frac{1}{9} \frac{1}{9} \frac{1}{9} \]
Since there's no need to further simplify, we can express the entire repeating decimal as:
\[ \frac{1}{9} + \frac{1}{9} + \frac{1}{9} = \frac{1}{9} \cdot 3 = \frac{3}{9} \]
Thus, the infinite decimal expansion \( 0.111111\ldots 0.111111\ldots 0.111111\ldots \) can be expressed as the fraction:
\[ \frac{1}{9} \]
However, given that it consists of three segments of \( 0.111111\ldots \), you can also simply write it as:
\[ \frac{1}{9} + \frac{1}{9} + \frac{1}{9} = \frac{3}{9} \]
So, the final fraction representation of the infinite decimal \( 0.111111\ldots 0.111111\ldots 0.111111\ldots \) is:
\[ \frac{1}{9} \text{ (each segment) or roughly } \frac{3}{9}. \]
To match the displayed format clearly, the final result is:
\[ \frac{1}{9} \] (Note: You can state that the infinite decimal is equivalent to \(\frac{3}{9}\), but \(\frac{1}{9}\) is the value of each individual segment).