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The weighted binomial sum $f_m(r)=2^{-r}\sum_{i=0}^r\binom{m}{i}$ arises incoding theory and information theory. We prove that,for $m\not\in\{0,3,6,9,12\}$, the maximum value of $f_m(r)$ with $0\leqslant r\leqslantm$ occurs when $r=\lfloor m/3\rfloor+1$. We also show this maximum value isasymptotic to $\frac{3}{\sqrt{{\pi}m}}\left(\frac{3}{2}\right)^m$ as$m\to\infty$.

On the maximum of the weighted binomial sum $2^{-r}\sum_{i=0}^r\binom{m}{i}$