# How to show that <munderover> <mo movablelimits="false">&#x2211;<!-- ∑ --> <mrow class="M

How to show that $\sum _{m=1}^{n/4}\left(\genfrac{}{}{0}{}{n}{m}\right){2}^{-n}\le \mathrm{exp}\left(-n/8\right)$
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Ordettyreomqu
For convenience, lets replace $n\to 4n$, so that it is equivalent to show $\sum _{m=1}^{n}\left(\genfrac{}{}{0}{}{4n}{m}\right)⩽{2}^{4n}{e}^{-n/2}$. We need a good bound for the partial sum in LHS, so consider

As $1+\frac{1}{3}+\frac{1}{{3}^{2}}+\cdots =\frac{3}{2}$, we get the bound $\sum _{m=1}^{n}\left(\genfrac{}{}{0}{}{4n}{m}\right)<\frac{3}{2}\left(\genfrac{}{}{0}{}{4n}{n}\right)$ for the partial sum. Now, using an easily found and rather well known upper bound for binomial coefficients, viz. $\left(\genfrac{}{}{0}{}{n}{k}\right)<\frac{{n}^{k}}{k!}$, it is enough to show
$\frac{3}{2}\frac{\left(4n{\right)}^{n}}{n!}<{2}^{4n}\cdot {e}^{-n/2}$
which is straightforward by induction.