Editing Summation (section)

== Notation ==
{{Further information|Iterated binary operation#Notation}}
=== Capital-sigma notation ===
[[File:Sigma summation notation.svg|thumb|An explanation of the sigma (Σ) summation notation|class=skin-invert-image]]
Mathematical notation uses a symbol that compactly represents summation of many similar terms: the ''summation symbol'', <math display="inline">\sum</math>, an enlarged form of the upright capital Greek letter [[sigma]].<ref>{{cite book
 | last = Apostol | first = Tom M.
 | title = Calculus
 | year = 1967
 | publisher = [[John Wiley & Sons]]
 | isbn = 0-471-00005-1
 | edition = 2nd
 | volume = 1
 | location = USA
 | pages = 37
}}</ref> This is defined as
<math display="block">\sum_{i \mathop =m}^n a_i = a_m + a_{m+1} + a_{m+2} + \cdots + a_{n-1} + a_n</math>
where {{math|''i''}} is the "index of summation" or "dummy variable"{{sfnp|Koshy|2002|p=[https://books.google.com/books?id=-9pg-4Pa19IC&pg=PA10 10]}}, {{math|''a<sub>i</sub>''}} is an indexed variable representing each term of the sum; {{math|''m''}} is the "lower bound of summation", and {{math|''n''}} is the "upper bound of summation". The "{{math|1=''i'' = ''m''}}" under the summation symbol means that the index {{math|''i''}} starts out equal to {{math|''m''}}. The index, {{math|''i''}}, is incremented by one for each successive term, stopping when {{math|1=''i'' = ''n''}}.{{efn|For a detailed exposition on summation notation, and arithmetic with sums, see {{cite book
 | last1 = Graham | first1 = Ronald L.
 | last2 = Knuth | first2 = Donald E.
 | last3 = Patashnik | first3 = Oren
 | year = 1994
 | title = Concrete Mathematics: A Foundation for Computer Science
 | edition = 2nd
 | chapter = Chapter 2: Sums
 | publisher = Addison-Wesley Professional
 | isbn = 978-0201558029
}}}} This is read as "sum of {{math|''a<sub>i</sub>''}}, from {{math|1=''i'' = ''m''}} to {{math|''n''}}". However, some notations may include the index at the upper bound of summation, or omit the indec at the lower bound as in <math display="inline"> \sum_{i=m} ^{i=n} a_i </math> or <math display="inline"> \sum_m ^n a_i </math>, respectively.{{sfnp|Koshy|2002|p=[https://books.google.com/books?id=-9pg-4Pa19IC&pg=PA9 9]}} In some cases, there are sigma notation where the range of bounds is omitted, which denotes the dummy variable only, like <math display="inline"> \sum_i a_i </math>.{{sfnp|Vivaldi|2014|p=[https://books.google.com/books?id=wpQvBQAAQBAJ&pg=PA34 34]}} Here is an example showing the summation of squares:
<math display="block">\sum_{i = 3}^6 i^2 = 3^2+4^2+5^2+6^2 = 86.</math>
In general, while any variable can be used as the index of summation (provided that no ambiguity is incurred), some of the most common ones include letters such as <math>i</math>,{{efn|In contexts where there is no possibility of confusion with the [[imaginary unit]] <math>i</math>}} <math>j</math>, <math>k</math>, and <math>n</math>; the latter is also often used for the upper bound of a summation.<ref name="franco"/> Alternatively, the index and bounds of summation are sometimes omitted from the definition of summation if the context is sufficiently clear. This applies particularly when the index runs from 1 to ''n''. For example, one might write that <math display="inline">\sum a_i = \sum_{i = 1}^n a_i</math>.<ref>{{Cite web|title=Summation Notation|url=http://www.columbia.edu/itc/sipa/math/summation.html|access-date=2020-08-16|website=www.columbia.edu}}</ref>

Generalizations of this notation are often used, in which an arbitrary logical condition is supplied, and the sum is intended to be taken over all values satisfying the condition. For example, <math display="inline">\sum_{0 \le k < 100} f(k)</math>
is an alternative notation for  <math display=inline>\sum_{k = 0}^{99} f(k),</math> the sum of <math>f(k)</math> over all ([[integer]]s) <math>k</math> in the specified range.<ref name="franco">{{cite book
 | title = Mathematical Writing
 | first = Franco | last = Vivaldi
 | url = https://books.google.com/books?id=wpQvBQAAQBAJ&pg=PA35
 | page = 35
 | publisher = Springer
 | doi = 10.1007/978-1-4471-6527-9
 | year = 2014
}}</ref> Similarly, <math display="inline">\sum_{x \mathop \in S} f(x)</math> is the sum of <math>f(x)</math> over all elements <math>x</math> in the set <math>S</math>,<ref>{{cite book
 | last = Miller | first = Victor S.
 | title = Handbook of Discrete and Combinatorial Mathematics
 | chapter = Finite Sums and Summation
 | editor-first = Kenneth H. | editor-last = Rosen
 | url = http://books.google.com/books?id=Xj4PEAAAQBAJ&pg=PA196
 | page = 196
}}</ref><ref>{{cite book
 | title = Elementary Number Theory with Applications
 | first = Thomas | last = Koshy
 | year = 2002
 | url = https://books.google.com/books?id=-9pg-4Pa19IC&pg=PA12
 | page = 12
 | publisher = [[Harcourt (publisher)|Harcourt]]
}}</ref> and <math display="inline">\sum_{d\,|\,n}\;\mu(d)</math>
is the sum of <math>\mu(d)</math> over all positive integers <math>d</math> [[divisor|dividing]] <math>n</math>.{{efn|Although the name of the [[Free variables and bound variables|dummy variable]] does not matter (by definition), one usually uses letters from the middle of the alphabet (<math>i</math> through <math>q</math>) to denote integers, if there is a risk of confusion. For example, even if there should be no doubt about the interpretation, it could look slightly confusing to many mathematicians to see <math>x</math> instead of <math>k</math> in the above formulae involving <math>k</math>.}}

There are also ways to generalize the use of many sigma notations. For example, one writes double summation as two sigma notations with different dummy variables <math display="inline"> \sum_{i=\ell}^n \sum_{j=m}^k a_{i,j} </math>. Considering that the both sigma notation's range are the same, the double sigma notations can be wrapped into a single notation, so the double summation is rewritten as <math display="inline">\sum_{i=m}^n \sum_{j=m}^n a_{i,j} = \sum_{i,j=m}^n a_{i,j}</math>.{{sfnp|Vivaldi|2014|p=[https://books.google.com/books?id=wpQvBQAAQBAJ&pg=PA36 36]}}

The term '''{{vanchor|finite series}}''' is sometimes used when discussing the summation presented above. Contrast to the [[Series (mathematics)|infinite series]], the upper bound tends to [[infinity]] <math display="inline"> \sum_{i=m}^\infty a_i </math>, which results in converge if there is a result of the sum, or diverge if otherwise. The bound in the infinite series's sigma notation can be alternatively denoted as <math display="inline"> \sum_{i \ge 0} a_i </math>.{{sfnp|Vivaldi|2014|p=[https://books.google.com/books?id=wpQvBQAAQBAJ&pg=PA36 36]}}

Relatedly, the similar notation is used for the [[product of a sequence]], where <math display="inline">\prod</math>, an enlarged form of the Greek capital letter [[Pi (letter)|pi]], is used instead of <math display="inline">\sum</math>.{{sfnp|Koshy|2002|p=[https://books.google.com/books?id=-9pg-4Pa19IC&pg=PA13 13]}}

===Special cases===
It is possible to sum fewer than 2 numbers:
* If the summation has one summand <math>x</math>, then the evaluated sum is <math>x</math>.
* If the summation has no summands, then the evaluated sum is [[0 (number)|zero]], because zero is the [[identity element|identity]] for addition. This is known as the ''[[empty sum]]''.

These degenerate cases are usually only used when the summation notation gives a degenerate result in a special case.
For example, if <math>n=m</math> in the definition above, then there is only one term in the sum; if <math>n=m-1</math>, then there is none.

===Algebraic sum===
The phrase 'algebraic sum' refers to a sum of terms which may have positive or negative signs. Terms with positive signs are added, while terms with negative signs are subtracted. e.g.
+1 −1

=== 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]}}