Volumes of polytopes via power series

Hopefully we know that $\displaystyle \sum_{i \geq 0} x^i $ $= \frac{1}{1-x}$. Similarly one computes that $ \displaystyle \sum_{i \leq 1} x^i $ $= \frac{x^2}{x-1}$. Interestingly, $ \frac{1}{1-x} + \frac{x^2}{x-1} = 1 + x$ which is the sum corresponding to the integers in the interval $[0, 1] = [0, \infty] \cap [-\infty, 1]$. We will explain generalization of this (called Brion's theorem) to integer points in convex polytopes of arbitrary dimension. Surprisingly, this gives a formula for the volume of a polytope in terms of summing up certain rational functions associated to vertices of the polytope. We also discuss related theorems of Lawrence-Varchenko and Brianchon-Gram about characteristic function of a convex polytope.