Sum of Infinite Geometric Sequence/Corollary 2

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Corollary to Sum of Infinite Geometric Sequence

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:

$\displaystyle \sum_{n \mathop = 0}^\infty a z^n = \frac a {1 - z}$


Proof

\(\ds \sum_{n \mathop = 0}^\infty a z^n\) \(=\) \(\ds a \sum_{n \mathop = 0}^\infty z^n\)
\(\ds \) \(=\) \(\ds a \frac 1 {1 - z}\) Sum of Infinite Geometric Sequence
\(\ds \) \(=\) \(\ds \frac a {1 - z}\)

$\blacksquare$


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