TMUA 2023 Paper 2 Question 15

The base $10$ number $0.03841$ has the value $$0\times 10^{-1}+3\times 10^{-2}+8\times 10^{-3}+4\times 10^{-4}+1\times 10^{-5}=\mathrm{0.03841.}$$ Similarly, the base $2$ number $0.01101$ has the value $$0\times 2^{-1}+1\times 2^{-2}+1\times 2^{-3}+0\times 2^{-4}+1\times 2^{-5}=\frac{13}{32}.$$ What is the value of the recurring base $2$ number $$0.\dot{0}01\dot{1}=0.001100110011\dots ?$$