Is there any way to approximate the following sum:
$$ \sum\limits_{i_1=1}^N\sum\limits_{i_1=2}^N \cdots \sum\limits_{i_k=1}^N \cdots\sum\limits_{i_N=1}^N \exp(-r_{i_1}-r_{i_{k+1}}-r_{i_{2k+1}}- r_{i_{3k+1}}\cdots -r_{i_N}) $$
where $k<<N$, $i_1 \neq i_2 \neq i_k \cdots \neq i_N$ and $r_{i_j}$ is any positive variable. Final expression could be in terms of $r_{i_j}$. Please explain and thanks.
Since \begin{align} \sum_{k=1}^{m} x^{k} = -1 + \frac{1-x^{m+1}}{1-x} = \frac{x(1-x^{m})}{1-x} \end{align} then \begin{align} \sum_{k=1}^{m} e^{-k} = \frac{1-e^{-m}}{e-1}. \end{align} Each sum that does not have an index of the summed function provides the value $N$. As the summation appears \begin{align} \sum_{i_1=1}^{N} \sum_{i_2=1}^{N} \cdots \sum_{i_k=1}^{N} \cdots\sum_{i_N=1}^{N} e^{-i_1-i_{k+1}-i_{2k+1}- \cdots - i_{N}} \end{align} the result is \begin{align} N^{2k} \left( \frac{1-e^{-N}}{e-1} \right)^{N-2k}. \end{align} If the summation is supposed to read \begin{align} \sum_{i_1=1}^{N} \sum_{i_2=1}^{N} \cdots \sum_{i_k=1}^{N} \cdots\sum_{i_N=1}^{N} e^{-i_1 - \cdots - i_{k} - \cdots - i_{N}} \end{align} then the value is \begin{align} \left( \frac{1-e^{-N}}{e-1} \right)^{N}. \end{align}