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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