Suppose that \begin{equation} S=\sum\limits_{k=0}^{N} (-1)^k \exp(-c_k) \end{equation} where $c_k$ are known and positive.
Given that \begin{equation} \sum\limits_{k=0}^{N} (-1)^k c_k=0 \end{equation} is there a way of simplifying the expression of $S$?
Thank you