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?

$$(1+x+x^2+x^3+x^4+\cdots)(1-x)=1.$$
