I was stuck by a problem for quite a long time, which is relating to the summation with binomial coefficients and fractions as below:
$\sum_{i=1}^M\sum_{j=1}^K(-1)^{i+j}\binom{M}{i}\binom{K}{j}\frac{ia}{ia+jb}e^{-\frac{ia+jb}{\rho}}$,
where $a$ and $b$ are constants and $\rho$ is approaching infinity.
I have tried to simplify the expression with approximation $e^{-\frac{1}{x}}\overset{\rho\rightarrow\infty}{\approx} 1-\frac{1}{x}$, or calculate the summation of $i$ at first then that of $j$. Unfortunately, my attempt has not been successful. Actually I would like to attain the form like $(1-e^{-a/\rho})^M(1-e^{-b/\rho})^K$, but I am not sure if this can be found or not.
I am really appreciated it if you can give me some insight into this problem. Many thanks in advance.