This question was inspired by the following post - "Finite Summation of Fractional Factorial Series" $$ $$
We know already that $$e^x=\frac{x^0}{0!}+\frac{x^1}{1!}+...$$
Suppose we want to approximate the sum of a well-defined subset of terms,
Can we (1) Pick out a limited 'run' of terms in $e^x$ such as $$\frac{x^3}{3!}+\frac{x^4}{4!}+\frac{x^5}{5!}$$ and (2) Can we, "sieve" those terms to get this kind of thing$$\frac{x^3}{3!}+\frac{x^5}{5!}+\frac{x^7}{7!}$$In this case, the distance is $2=(7-5)=(5-3)$, $1^{st}$ term index is 3, count of terms is 3.
For your first question note that $$ \sum_{n=k}^\infty \frac{x^n}{n!}=\exp(x)-x-\frac{x^2}{2!}-\dotsb-\frac{x^{k-1}}{k!} $$ for $k\geq 1$.