Is this a well-defined generating function? $$\frac{1}{x(1+x)}$$
We know that $\:\frac{1}{(1+x)} = \sum_{n \ge0}(-1)^nx^n$,$\:$ hence the notation $\frac{1}{x}\sum_{n \ge0}(-1)^nx^n \:$ would act as a shift for the coefficients $f(n) = (-1)^n$ to the prior power of $x$, however $\:\frac{1}{(1+x)}$ has the constant term.
I found it in an example and probably I must have misinterpreted something