Editing Summation (section)

=== History ===
The origin of the summation notation dates back to 1675 when [[Gottfried Wilhelm Leibniz]], in a letter to [[Henry Oldenburg]], suggested the symbol <math display="inline"> \int </math> to mark the sum of differentials ([[Latin]]: ''calculus summatorius''), hence the S-shape.<ref>{{cite book
 |first=David M. |last=Burton
 |title=The History of Mathematics: An Introduction
 |year=2011
 |edition=7th
 |publisher=McGraw-Hill
 |isbn=978-0-07-338315-6
 |page=414
}}</ref><ref>{{cite book
 |first=Gottfried Wilhelm |last=Leibniz |author-link=Gottfried Wilhelm Leibniz
 |title=Der Briefwechsel von Gottfried Wilhelm Leibniz mit Mathematikern. Erster Band
 |url=http://name.umdl.umich.edu/AAX2762.0001.001
 |year=1899
 |editor-first=Karl Immanuel
 |editor-last=Gerhardt
 |place=Berlin
 |publisher=Mayer & Müller
 |page=[https://quod.lib.umich.edu/u/umhistmath/aax2762.0001.001/185?page=root;size=100;view=image 154]
}}</ref>{{sfnp|Cajori|1929|pages=[https://archive.org/details/in.ernet.dli.2015.88254/page/n203 181-182]}} The renaming of this symbol to ''[[integral]]'' arose later in exchanges with [[Johann Bernoulli]].{{sfnp|Cajori|1929|pages=[https://archive.org/details/in.ernet.dli.2015.88254/page/n203 181-182]}} In 1755, the summation symbol Σ is attested in [[Leonhard Euler]]'s ''[[Institutiones calculi differentialis]]''.{{sfnp|Cajori|1929|p=[https://archive.org/details/in.ernet.dli.2015.88254/page/n83 61]}}<ref>{{cite book
 |last1=Euler |first1=Leonhard |author-link=Leonhard Euler
 |title=Institutiones Calculi differentialis
 |date=1755
 |location=Petropolis
 |page=[https://www.digitale-sammlungen.de/en/view/bsb10053431?page=54,55 27]
 |url=https://www.digitale-sammlungen.de/en/view/bsb10053431?page=54,55
 |language=Latin
}}</ref> Euler uses the symbol in expressions like <math display="inline"> \sum (2wx + w^2) = x^2</math>. The usage of sigma notation was later attested by mathematicians such as [[Lagrange]], who denoted <math display="inline"> \sum </math> and <math display="inline"> \sum ^n </math> in 1772.{{sfnp|Cajori|1929|p=[https://archive.org/details/in.ernet.dli.2015.88254/page/n83 61]}}<ref>{{cite book
 |last1=Lagrange
 |first1=Joseph-Louis
 |author-link=Joseph-Louis Lagrange
 |title=Oeuvres de Lagrange. Tome 3
 |date=1867–1892
 |location=Paris
 |page=[https://gallica.bnf.fr/ark:/12148/bpt6k229222d/f452.item 451]
 |url=https://gallica.bnf.fr/ark:/12148/bpt6k229222d/f452.item
 |language=French
}}</ref> [[Joseph Fourier|Fourier]] and [[Carl Gustav Jacob Jacobi|C. G. J. Jacobi]] also denoted the sigma notation in 1829,{{sfnp|Cajori|1929|p=[https://archive.org/details/in.ernet.dli.2015.88254/page/n83 61]}} but Fourier included lower and upper bounds as in <math display="inline">\sum_{i=1}^{\infty}e^{-i^2t} \ldots</math>.<ref>{{cite book
 |title=Mémoires de l'Académie royale des sciences de l'Institut de France pour l'année 1825, tome VIII
 |date=1829
 |publisher=Didot
 |location=Paris
 |pages=[https://books.google.com/books?id=Mpu9XDBOmagC&pg=583 581-622]
 |url=https://books.google.com/books?id=Mpu9XDBOmagC&pg=583
 |language=French
}}</ref><ref>{{cite book
 |last1=Fourier
 |first1=Jean-Baptiste Joseph
 |author-link=Joseph Fourier
 |title=Oeuvres de Fourier. Tome 2
 |date=1888–1890
 |publisher=Gauthier-Villars
 |location=Paris
 |page=[https://gallica.bnf.fr/ark:/12148/bpt6k33707/f154.item 149]
 |url=https://gallica.bnf.fr/ark:/12148/bpt6k33707/f154.item
 |language=French
}}</ref> Other than sigma notation, the capital letter ''S'' is attested as a summation symbol for series in 1823, which was apparently widespread.{{sfnp|Cajori|1929|p=[https://archive.org/details/in.ernet.dli.2015.88254/page/n83 61]}}