I opened Stanley's Enumerative combinatorics Vol. 1 and looked at example 1.1.11:
Suppose $F(0)=1$ and let $G(x)$ be the unique power series satisfying
$G'(x)=F'(x)/F(x)$, $G(0)=0$.
The author wants to show the equivalence between this and the expression $F(x)=\exp G(x)$ of formal power series.
He lets $F(x)=1+\sum_{n \geq 1}a_nx^n$, computes the coefficients of $G(x)$ and $\exp G(x)$, and concludes that the latter must be polynomials in the $a_n$'s. He then states that the coefficients of $F(x)$ and $\exp G(x)$ must agree by invoking a uniqueness theorem of Taylor.
Could someone please elaborate on the authors intention or the point(s) he is trying to make, and perhaps give the proof with a bit of clarification or refer to an alternative. I was trying to let go of the analyst approach, but here the author seems to want to pass from the continuous setting to the extended one (judging by his "alternative method" that follows the one just given) by various forms of "function-theoretic" means, invoking various forms of convergence.