I have $K=e^{n\beta}$ positive numbers $D_1\leq D_2\leq ...\leq D_K$. I want to find the maximum value of $\beta$ for which
$e^{-nD_1}+e^{-nD_2}+...+e^{-nD_K}$
goes to zero as $n$ goes to infinity.
Also I know $\beta$ must be less than mean of $D_i$'s (i.e. $\beta<\frac{\sum_{i=1}^K D_i}{K}$) and $\beta<D_1$ is not the best I can do. Can anyone please help me?!