Vrftsjtryk

In my discrete math class, we are studying generating functions. We learned that
$$
e^x = sum_{i = 0}^{infty} frac{x^i}{i!},
$$
which is certainly an identity in calculus. However, in the ring of formal power series, is there any sort of concept of exponentiation? Suppose we have $F, G in R[[X]].$ Is there any sort of a concept or definition for
$$
F^G?
$$
I am mostly looking for a reason why the formal power series $e$ (i.e., the power series whose constant coefficient is $e$ and all remaining coefficients are 0), when raised to the power of the power series $x$, satisfies the above identity. A generalization would be appreciated as well, for the $F^G$ case.

Look for the idea of substitution of formal power series. This is called composition in this Wikipedia article.
To be more specific with respect to $F^G$, this can be done for example in the case $G=sum_{i=1}^infty g_i X^i$ and $F=1+H$ with $H=sum_{i=1}^infty h_i X^i$ by considering the formal power series $exp(G ln(1+H))$ where $ln(1+H):=sum_{i=1}^infty (-1)^{i-1}frac{H^i}{i}$.