What properties of the exponential function are valid because of formal power series properties?

Question

To what extent the formal power series $\sum_{n=0}^{\infty}\frac{X^{n}}{n!}$ has nice properties without assuming any convergence concepts, just the elementary formal power series machinery? I think the title summarize my question the best. In particular, are the usual properties of the real and complex trigonometric, logarithmic and exponential functions always calculus/analysis/group theory theorems or in fact some of them are formally valid (formal power series formalism)? I know this previous question that concerns only one of those properties (the multiplicative functional equation property).

Answer 1

The differential equation for $\exp(x) = \mathrm{d}/\mathrm{d}x(\exp(x))$ follows from the formal property of formal power series and formal derivatives (in the sense that you define $D(x^n) := nx^{n-1}$ for polynomials and extend this to formal power series by termwise formal differentiation). Then you get completely formally $$ D\left(\exp(x)\right) = D\left(\sum_{n=0}^{\infty}\frac{x^n}{n!}\right) \overset{!}{=} \sum_{n=0}^{\infty} D\left(\frac{x^n}{n!}\right) = \sum_{n=1}^{\infty}\frac{nx^{n-1}}{n!} = \sum_{n=1}^{\infty}\frac{x^{n-1}}{(n-1)!} = \sum_{n=0}^{\infty}\frac{x^n}{n!}. $$ I've identified the step where I used the formal derivative of power series with a $\overset{!}{=}$ symbol, as this is where you need to argue something about the uniform convergence of $\exp(x)$ to know that step is legit in analysis.

For what it's worth, this is the kind of argument you'd see with generating function technology as well.