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:

$\ds \dfrac 1 {1 - z} \map G z = \sum_{n \mathop \ge 0} c_n z^n$

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

$\ds \sequence {c_n}$

$=$

$\ds a_0 + \paren {a_0 + a_1} + \paren {a_0 + a_1 + a_2} + \cdots$

$\ds $

$=$

$\ds \sum_{n \mathop \ge 0} \paren {\sum_{k \mathop = 0}^n a_k}$

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 $\sequence {b_n}$.

$\map H z = \dfrac 1 {1 - z}$

Then:

$\ds \map G z \map H z$

$=$

$\ds \sum_{n \mathop \ge 0} \paren {\sum_{k \mathop = 0}^n a_k b_{n - k} } z^n$

$\ds \leadsto \ \ $

$\ds \dfrac 1 {1 - z} \map G z$

$=$

$\ds \sum_{n \mathop \ge 0} \paren {\sum_{k \mathop = 0}^n a_k} z^n$

as all $b_{n - k} = 1$

$\ds $

$=$

$\ds \sum_{n \mathop \ge 0} c_n z^n$

Hence the result by definition of generating function.

$\blacksquare$