Generating Function for Sequence of Partial Sums of Series

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

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

Let $\sequence {c_n}$ denote the sequence of partial sums of $s$.

Then the generating function for $\sequence {c_n}$ is given by:
 * $\ds \dfrac 1 {1 - z} \map G z = \sum_{n \mathop \ge 0} c_n z^n$

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

Consider the sequence $\sequence {b_n}$ defined as:


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

Let $\map H z$ be the generating function for $\sequence {b_n}$.

By Generating Function for Constant Sequence:
 * $\map H z = \dfrac 1 {1 - z}$

Then:

Hence the result by definition of generating function.