Is the degree function well behaved over power series? For non-zero formal polynomials $x$ and $y$ it holds that $\deg(xy)=\deg x + \deg y$. Allowing for infinite degrees, does this formula hold for arbitrary non-zero power series? And is there a definition that extends $\deg$ such that the formula holds for arbitrary non-zero Laurent series?
 A: $$(1+x+x^2+x^3+x^4+\cdots)(1-x)=1.$$
A: A not so satisfying answer: $-\deg$ is a valuation on the field $K(t)$ of rational functions in the variable $t$. $K(t)$ is a subfield of the field $K((t))$ of formal Laurent series. Hence the valuation $-\deg$ can be extended to a valuation of $K((t))$ (in fact there are infinitely many such extensions) - in particular the formula under discussion holds for that extension. However, as far as I know these extensions do not have an obvious interpretation.
A: For formal power series the degree is not defined. If as you say you allow for infinity as value, then almost all series will have infinity as degree, which isn't very interesting, and even then the additive law fails for the few cases that remain. More interesting is to take the smallest degree of any (nonzero) term, which defines the valuation of a formal power series ring (the valuation is $+\infty$ for the zero series, only). The additive property of the valuation holds.
For instance the valuation can be used to prove that formal power series rings are integral domains (if the base ring is), replacing the degree in the proof of the similar statement for polynomial rings. (Note that you don't need to use the degree for polynomial rings here; the valuation will do just as fine there too). However the most importatnt use of the valuation is for topological purposes, defining convergence of infinite sums and such (a sum of formal power series converges if and only if the valuation of its terms tends to infinity).
