Editing Summation (section)

{{Short description|Addition of several numbers or other values}}
{{About|sums of several elements|more elementary aspects|Addition|infinite sums|Series (mathematics)|other uses}}

In [[mathematics]], '''summation''' is the [[addition]] of a [[sequence]] of [[number]]s, called ''addends'' or ''summands''; the result is their ''sum'' or ''total''. Beside numbers, other types of values can be summed as well: [[function (mathematics)|functions]], [[vector space|vectors]], [[matrix (mathematics)|matrices]], [[polynomial]]s and, in general, elements of any type of [[mathematical object]]s on which an [[operation (mathematics)|operation]] denoted "+" is defined.

Summations of [[infinite sequence]]s are called [[series (mathematics)|series]]. They involve the concept of [[limit (mathematics)|limit]], and are not considered in this article.

The summation of an explicit sequence is denoted as a succession of additions. For example, summation of {{math|[1, 2, 4, 2]}} is denoted {{math|1 + 2 + 4 + 2}}, and results in 9, that is, {{math|1=1 + 2 + 4 + 2 = 9}}. Because addition is [[associative]] and [[commutative]], there is no need for parentheses, and the result is the same irrespective of the order of the summands. Summation of a sequence of only one summand results in the summand itself. Summation of an empty sequence (a sequence with no elements), by convention, results in 0.

Very often, the elements of a sequence are defined, through a regular pattern, as a [[function (mathematics)|function]] of their place in the sequence. For simple patterns, summation of long sequences may be represented with most summands replaced by ellipses. For example, summation of the first 100 [[natural number]]s may be written as {{math|1 + 2 + 3 + 4 + ⋯ + 99 + 100}}. Otherwise, summation is denoted by using [[#Capital-sigma notation|Σ notation]], where <math display="inline">\sum</math> is an enlarged capital [[Greek letter]] [[sigma]]. For example, the sum of the first {{mvar|n}} natural numbers can be denoted as

:<math>\sum_{i=1}^n i</math>

For long summations, and summations of variable length (defined with ellipses or Σ notation), it is a common problem to find [[closed-form expression]]s for the result. For example,{{efn|For details, see [[Triangular number]].}}

:<math>\sum_{i=1}^n i = \frac{n(n+1)}{2}.</math>

Although such formulas do not always exist, many summation formulas have been discovered—with some of the most common and elementary ones being listed in the remainder of this article.