Laplace Transform of Generating Function of Sequence

Theorem
Let $\sequence{a_n}$ be a sequence which has a generating function which is convergent.

Let $\map G z$ be the generating function for $\sequence{a_n}$.

Let $\map f x$ be the step function:
 * $\map f x = \ds \sum_{k \mathop \in \Z} a_k \sqbrk{0 \le k \le x}$

where $\sqbrk{0 \le k \le x}$ is Iverson's convention.

Then the Laplace transform of $\map f x$ is given by:
 * $\laptrans {\map f s} = \dfrac {\map G {e^{-s} }} s$

Proof
We note that:

and:

Then: