Editing Summation (section)

=== 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]}}