Multiplicative inverse of infinite series- polynomial

Question

I believe there's a multiplicative inverse for the set of the series $\sum_{i=-\infty}^\infty x^i$. If this is the case, how would I go about finding the multiplicative inverse? Otherwise, is there just not a multiplicative inverse?

Answer 1

Let $S:=\sum_{i=-\infty}^\infty x^i$. You probably want $xS=S$ to hold, and this gives $(x-1)S=0$ which leads to $S=0$ and has no multiplicative inverse. Of course, this is assuming that the definition of $S$ makes sense. You need to specify a summation method for that because it diverges for all $x$.