Generating Function for Sequence of Partial Sums of Series

Theorem
Let $s$ be the the series:
 * $\displaystyle s = \sum_{n \mathop = 1}^\infty a_n = a_0 + a_1 + a_2 + a_3 + \cdots$

Let $G \left({z}\right)$ be the generating function for the sequence $\left\langle{a_n}\right\rangle$.

Let $\left\langle{c_n}\right\rangle$ denote the sequence of partial sums of $s$.

Then the generating function for $\left\langle{c_n}\right\rangle$ is given by:
 * $\displaystyle \dfrac 1 {1 - z} G \left({z}\right) = \sum_{n \mathop \ge 0} c_n z^n$

Proof
By definition of sequence of partial sums of $s$:

Consider the sequence $\left\langle{b_n}\right\rangle$ defined as:


 * $\forall n \in \Z_{\ge 0}: b_n = 1$

Let $H \left({z}\right)$ be the generating function for $\left \langle {b_n}\right \rangle$.

By Generating Function for Constant Sequence:
 * $H \left({z}\right) = \dfrac 1 {1 - z}$

Then:

Hence the result by definition of generating function.