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:
- $\ds \sum_{n \mathop = 0}^\infty a z^n = \frac a {1 - z}$
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 | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac a {1 - z}\) |
$\blacksquare$
Sources
- 1964: Milton Abramowitz and Irene A. Stegun: Handbook of Mathematical Functions ... (previous) ... (next): $3$: Elementary Analytic Methods: $3.1$ Binomial Theorem etc.: Sum of Geometric Progression to $n$ Terms: $3.1.10$
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 19$: Geometric Series: $19.5$
- 1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics ... (previous) ... (next): geometric series
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): geometric series
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): geometric series
- 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): geometric series