## § Ring of power series with infinite positive and negative terms

If we allow a ring with elements $x^i$ for all $-\infty < x < \infty$, for notation's sake, let's call it $R[[[x]]]$. Unfortunately, this is a badly behaved ring. Define $S \equiv \sum_{i = -\infty}^\infty x^i$. See that $xS = S$, since multiplying by $x$ shifts powers by 1. Since we are summing over all of $\mathbb Z$, $+1$ is an isomorphism. Rearranging gives $(x - 1)S = 0$. If we want our ring to be an integral domain, we are forced to accept that $S = 0$. In the Barvinok theory of polyhedral point counting, we accept that $S = 0$ and exploit this in our theory.