Sum of Infinite Geometric Sequence/Proof 4

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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 $\dfrac 1 {1 - z}$.


Proof

\(\ds \frac 1 {1 - z}\) \(=\) \(\ds \frac 1 {1 + \paren {-z} }\)
\(\ds \) \(=\) \(\ds \sum_{k \mathop = 0}^\infty \paren {-1}^k \paren {-z}^k\) Power Series Expansion of $\dfrac 1 {1 + z}$
\(\ds \) \(=\) \(\ds \sum_{k \mathop = 0}^\infty \paren {-1}^k \paren {-1}^k z^k\)
\(\ds \) \(=\) \(\ds \sum_{k \mathop = 0}^\infty \paren {-1}^{2 k} z^k\)
\(\ds \) \(=\) \(\ds \sum_{k \mathop = 0}^\infty z^k\)

$\blacksquare$


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