Exponential of a locally nilpotent infinitely powered endomorphism

Definition

Suppose $G$ is an abelian group and $f$ is a locally nilpotent endomorphism of $G$ that is also an 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.