# Convert binary to decimal

Calculus Level 5

Given a real number $$x\in[0; 1)$$ with binary expansion $$x=(0.a_1a_2a_3\ldots)_2$$, let $$f(x)=(0.a_1a_2a_3\ldots)_{10}$$ be the number obtained when interpreting the binary expansion of $$x$$ as a decimal expansion.

For example, $$f\left(\dfrac{1}{2}\right)=f((0.100\ldots)_2)=(0.100\ldots)_{10}=\dfrac{1}{10}$$.

Given that: $$\displaystyle I=\int\limits_0^1 f(x)dx=\dfrac{m}{n}$$, where $$m,n$$ are coprime positive integers.

Find the value of $$m+n$$.

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