Exponential of a locally nilpotent infinitely powered endomorphism

Definition
Suppose $$G$$ is an abelian group and $$f$$ is a defining ingredient::locally nilpotent endomorphism of $$G$$ that is also an defining ingredient::infinitely powered endomorphism. In other words, for every $$x \in G$$, there exists $$n \in \mathbb{N}$$ such that $$f^n(x) = 0$$. Further, for all natural numbers $$n$$, the image $$f^n(G)$$ is powered for all primes less than or equal to $$n$$. Then, the exponential of $$f$$ is the function:

$$x \mapsto x + f(x) + \frac{f(f(x))}{2!} + \frac{f(f(f(x)))}{3!} + \dots + \frac{f^{(n-1)}(x)}{(n - 1)!}, \mbox{ where } f^n(x) = 0$$

In other words, we add terms of the exponential series till they become zero.