In every source I've read about generating functions if $A(x)$ is a generating function, we treat $x$ as an indeterminate, in particular, it is treated as a placeholder. And if one goes on with what Wikipedia says about an indeterminate variable, https://en.wikipedia.org/wiki/Indeterminate_(variable), one can not treat $x$ as a number, for instance (i might be misunderstanding).
I am having the confusion about why in some cases we can assign a value to $x$. I'll appreciate if someone clarifies my misunderstanding.
Example, in Stanley's Enumerative Combinatorics he makes x=1 when trying to prove that $$n2^{n-1} = \sum_{k\geq0}k\binom{n}{k} $$
Every formal power series $f(x)=\sum_{n\ge 0} a_nx^n$ has a radius of convergence, $R$, which is can be found via $$ 1/R = \limsup_{n\to\infty} \sqrt[n]{a_n}. $$ If the $\limsup$ is zero, then $R$ is defined to be $\infty$. Now, the rule is this:
In particular, if $f(x)$ is a polynomial, then the $R$ is always $\infty$, so you can substitute anything for $x$. This is the case for the Stanley example in your post, the polynomial being $n(1+x)^{n-1}$.