Infinite Geometric Series as a Limit

Using an infinite geometric series, what is the binary value $0.1010101\dots$ equal to in base 10? Express your answer as a fraction in lowest terms.

(Hint: Non-integers in binary can be confusing. Just remember that each place represents a power of $2$. For example, $0.1$ in binary is $\frac{1}{2}$ in base 10, $0.01$ in binary is $\frac{1}{4}$ in base 10, $0.001$ in binary is $\frac{1}{8}$ in base 10, and so on.)