When talking about say exponential generating functions, there are a couple ways to think of it that I'm aware of.
The first is to think about it as formal series, denote this way by (1).
The latter is to think of it as analytic functions over some disk (if there is such disk with radius larger than 0, denoted by way (2).
The sweet thing about (2), when we assume all functions we're dealing with at the moment converge in a disk of radius more than 0 is that we can do freely without worry: we can take derivatives, multiply and add as we wish, and the result will be what we want because of the uniqueness of expansion as power series.
The disadvantage of (2) is clear, sometimes we are counting objects who's coefficients grow too large. This is where (1) has a benefit. However, the problem with (1) is that I don't feel comfortable regarding some things: Say we have this e.g.f, $F(x)$, and we have determined that $F(x)=1+xF(x)^2$ (as formal power series, with the operations defined on formal sums), in examples I see we then solve the function not in a formal sense, like here they'd say $F(x)=\frac{1-\sqrt{1-4x}}{2x}$, and then find the taylor expansion and thereby determine the coefficients. There's no way this is possible by (1), the case must be that they assume apriori that for those power series there will a good radius of convergence and like with ODE's, if it works perfect, that's the thinking right?
I'm not completly satisfied with this, what if we want to solve another functional equation as above but in which case the coefficients grow too fast for there to be a radius of convergance? How would we overcome it? Or come to tihnk of it, is there a way to say that the mere existence of the functional equations assures us the coefficients don't grow too fast? I think I'm digging too deep at something that should be simpler.