The value of $0.3333\ldots$ (where the "3" repeats indefinitely) is a repeating decimal that can be represented as a fraction.

Let's denote $x = 0.3333\ldots$.

Step-by-Step Solution:

1. Set up the equation:

   $x = 0.3333\ldots$

2. Multiply both sides by 10:

   $10x = 3.3333\ldots$

3. Subtract the original equation from this new equation:

   $10x - x = 3.3333\ldots - 0.3333\ldots$

   This simplifies to:

   $9x = 3$

4. Solve for $x$:

   $x = \frac{3}{9} = \frac{1}{3}$

Conclusion:

The value of $0.3333\ldots$ is $\frac{1}{3}$.

Would you like more details or have any questions? Here are some related topics you might find interesting:

1. How to convert other repeating decimals to fractions.
2. Why repeating decimals are rational numbers.
3. The process of finding decimal expansions of fractions.
4. Understanding infinite series and their sums.
5. How to prove that $0.9999\ldots = 1$.

Tip: Any repeating decimal can be expressed as a fraction by setting it up as an equation and solving algebraically!