Asymptotic Summation of Slow Converging and Rapidly Oscillating Series

Mean values of some observables describing quantum interaction between the Bose field in a cavity and a movable mirror can be represented as expectations of rapidly oscillating functions w.r.t. the Poisson measure with a large mean value ($N\approx 10^{23}$) corresponding to the average number of photons in laser beam. Straightforward summation of the series is impossible because over $2\sqrt N$ summands make a significant contribution. We derive an analytical expression approximating this sum with the error $O(N^{-1})$.