Sum of Infinite Geometric Sequence/Proof 3

Theorem

Let $S$ be a standard number field, that is $\Q$, $\R$ or $\C$.

Let $z \in S$.

Let $\size z < 1$, where $\size z$ denotes:

the absolute value of $z$, for real and rational $z$

the complex modulus of $z$ for complex $z$.

Then $\ds \sum_{n \mathop = 0}^\infty z^n$ converges absolutely to $\dfrac 1 {1 - z}$.

Let $S = \ds \sum_{n \mathop = 0}^\infty z^n$.

Then:

$\ds z S$

$=$

$\ds z \sum_{n \mathop = 0}^\infty z^n$

$\ds $

$=$

$\ds \sum_{n \mathop = 0}^\infty z^{n + 1}$

$\ds $

$=$

$\ds \sum_{n \mathop = 1}^\infty z^n$

$\ds $

$=$

$\ds S - z^0$

$\ds $

$=$

$\ds S - 1$

$\ds \leadsto \ \ $

$\ds 1$

$=$

$\ds \paren {1 - z} S$

Hence the result.

$\blacksquare$

1997: Donald E. Knuth: The Art of Computer Programming: Volume 1: Fundamental Algorithms (3rd ed.) ... (previous) ... (next): $\S 1.2.9$: Generating Functions: $(5)$