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Approximation of the sum involving binary entropy function

The second order Taylor expansion of $H(p)$ may suffice for you. Expanding $H(p)$ around $p = 1/2$ it is not difficult to show that $$H(p)=1-\frac{(1-2 p)^2}{2 \ln 2}+O\left(\left(p-\frac{1}{2}\right) …
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