I am calculating a bound for a random variable. Using the Chernoff bound, I get to the following: $$ \sum_{s=1}^{\frac{n}{2}} n^{ -2s\big[ \frac{\log(s)-1}{\log (n)} \pm 2 o_n(1) \big]} $$ Can someone help me to find the asymptotic behavior of the sum?
P.S.: $o_n(1)$ goes to zero as $n \to \infty$.