Editing Geometric series (section)

== Background ==
2,500 years ago, Greek mathematicians believed that an infinitely long list of positive numbers must sum to infinity. Therefore, [[Zeno of Elea]] created a [[Zeno's paradoxes|paradox]], demonstrating as follows: in order to walk from one place to another, one must first walk half the distance there, and then half of the remaining distance, and half of that remaining distance, and so on, covering infinitely many intervals before arriving. In doing so, he partitioned a fixed distance into an infinitely long list of halved remaining distances, each with a length greater than zero. Zeno's paradox revealed to the Greeks that their assumption about an infinitely long list of positive numbers needing to add up to infinity was incorrect.{{r|riddie}}

{{multiple image
 | image1 = Euclid book9 prop35 mod.png
 | caption1 = Elements of Geometry, Book IX, Proposition 35. "If there is any multitude whatsoever of continually proportional numbers, and equal to the first is subtracted from the second and the last, then as the excess of the second to the first, so the excess of the last will be to all those before it."
 | image2 = Parabolic Segment Dissection.svg
 | caption2 = Archimedes' dissection of a parabolic segment into infinitely many triangles
 | total_width = 500
}}
Euclid's ''[[Euclid's Elements|Elements]]'' has the distinction of being the world's oldest continuously used mathematical textbook, and it includes a demonstration of the sum of finite geometric series in Book IX, Proposition 35, illustrated in an adjacent figure.{{r|heiberg}}

[[Archimedes]] in his ''[[The Quadrature of the Parabola]]'' used the sum of a geometric series to compute the area enclosed by a [[parabola]] and a straight line. Archimedes' theorem states that the total area under the parabola is {{sfrac|4|3}} of the area of the blue triangle. His method was to dissect the area into infinite triangles as shown in the adjacent figure.{{r|sd}} He determined that each green triangle has {{sfrac|1|8}} the area of the blue triangle, each yellow triangle has {{sfrac|1|8}} the area of a green triangle, and so forth. Assuming that the blue triangle has area 1, then, the total area is the sum of the infinite series
<math display="block">1 + 2\left(\frac{1}{8}\right) + 4\left(\frac{1}{8}\right)^2 + 8\left(\frac{1}{8}\right)^3 + \cdots.</math>
Here the first term represents the area of the blue triangle, the second term is the area of the two green triangles, the third term is the area of the four yellow triangles, and so on. Simplifying the fractions gives
<math display="block">1 + \frac{1}{4} + \frac{1}{16} + \frac{1}{64} + \cdots, </math>
a geometric series with common ratio <math>r = \tfrac14</math> and its sum is:{{r|sd}}
:<math>\frac{1}{1 -r}\ = \frac{1}{1 -\frac{1}{4}} = \frac{4}{3}.</math>

[[File:Oresme diagram of two dimensional geometric series.png|thumb|A two dimensional geometric series diagram Nicole Oresme used to determine that the infinite series <math>\tfrac12 + \tfrac24 + \tfrac38 + \tfrac4{16} + \tfrac5{32} + \tfrac6{64} + \tfrac7{128} + \cdots</math> converges to 2.]]
In addition to his elegantly simple proof of the divergence of the [[Harmonic series (mathematics)|harmonic series]], [[Nicole Oresme]]<ref>{{cite web |last1=Babb |first1=J |title=Mathematical Concepts and Proofs from Nicole Oresme: Using the History of Calculus to Teach Mathematics |url=https://core.ac.uk/download/pdf/144470649.pdf |archive-url=https://web.archive.org/web/20210527033047/https://core.ac.uk/download/pdf/144470649.pdf |archive-date=2021-05-27 |url-status=live |publisher=The Seventh International History, Philosophy and Science Teaching conference|ref=oresme_lecture_notes |location=Winnipeg |pages=11–12, 21 |year=2003}}</ref> proved that the [[arithmetico-geometric series]] known as Gabriel's Staircase,<ref name="Swain2018">{{cite journal |last1=Swain |first1=Stuart G. |year=2018 |title=Proof Without Words: Gabriel's Staircase |journal=Mathematics Magazine |volume=67 |issue=3 |pages=209 |doi=10.1080/0025570X.1994.11996214 |issn=0025-570X}}</ref>
<math display="block"> \frac{1}{2}+\frac{2}{4}+\frac{3}{8}+\frac{4}{16}+\frac{5}{32}+\frac{6}{64}+\frac{7}{128}+\cdots = 2.</math>
In the diagram for his geometric proof, similar to the adjacent diagram, shows a two-dimensional geometric series. The first dimension is horizontal, in the bottom row, representing the geometric series with initial value <math>a=\tfrac12</math> and common ratio <math>r=\tfrac12</math>
<math display="block">
 S = \frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\frac{1}{16}+\frac{1}{32}+\cdots = \frac{\frac{1}{2}}{1 - \frac{1}{2}} = 1.
</math>
The second dimension is vertical, where the bottom row is a new initial term <math>a = S</math> and each subsequent row above it shrinks according to the same common ratio <math>r=\tfrac12</math>, making another geometric series with sum <math>T </math>,
<math display="block">
\begin{align}
T &= S \left(1 + \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \cdots\right)\\
&= \frac{S}{1-r} = \frac{1}{1-\frac{1}{2}} = 2.
\end{align}
</math>
This approach generalizes usefully to higher dimensions, and that generalization is described below in {{slink||Connection to the power series}}.