The repeating decimal $0.0101010101\ldots$ can be expressed as a fraction. Let's solve it step by step.

Let: $x = 0.0101010101\ldots$

Since the decimal repeats every two digits (01), multiply both sides of the equation by 100 to shift the decimal point two places: $100x = 1.0101010101\ldots$

Now subtract the original equation from this one: $100x - x = (1.0101010101\ldots) - (0.0101010101\ldots)$ $99x = 1$

Now, solve for $x$: $x = \frac{1}{99}$

Thus, $0.0101010101\ldots = \frac{1}{99}$.

Do you want further details or have any questions?

Here are 5 related questions to expand this concept:

1. How do you convert a repeating decimal like $0.\overline{3}$ to a fraction?
2. What is the fraction form of $0.001001001\ldots$?
3. How do you convert a repeating decimal with more than two repeating digits into a fraction?
4. Can any repeating decimal be expressed as a fraction?
5. How would you convert a mixed repeating decimal like $0.12\overline{3}$?

Tip: Repeating decimals can always be expressed as fractions of integers!