Truncated exponential of an endomorphism

Definition
Suppose $$G$$ is an abelian group, $$f$$ is an endomorphism of $$G$$ and $$r$$ is a nonnegative integer. The truncated exponential of $$f$$ to degree $$r$$, denoted $$\exp_r(f)$$, is defined as the endomorphism:

$$\exp_r(f) = x \mapsto x + \frac{f(x)}{1!} + \frac{f(f(x))}{2!} + \dots + \frac{f^{(r)}(x)}{r!}$$