Summation

In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, summation is the <a href="/facts/Addition/apFWJBFB">addition</a> of a <a href="/facts/Sequence/gV2S4uqp">sequence</a> of <a href="/facts/Number/auQSAE2m">numbers</a>, called addends or summands; the result is their sum or total. Beside numbers, other types of values can be summed as well: <a href="/facts/Function_(mathematics)/CdReEKCh">functions</a>, <a href="/facts/Vector_space/M1pxMLx2">vectors</a>, <a href="/facts/Matrix_(mathematics)/qa8FI4ko">matrices</a>, <a href="/facts/Polynomial/Lzak8VVx">polynomials</a> and, in general, elements of any type of <a href="/facts/Mathematical_object/Juhyow9M">mathematical objects</a> on which an <a href="/facts/Operation_(mathematics)/Yplv602Q">operation</a> denoted "+" is defined.
Summations of <a href="/facts/Infinite_sequence/gV2S4uqp">infinite sequences</a> are called <a href="/facts/Series_(mathematics)/yDnlsgIe">series</a>. They involve the concept of <a href="/facts/Limit_(mathematics)/j8JBhj4e">limit</a>, and are not considered in this article.
The summation of an explicit sequence is denoted as a succession of additions. For example, summation of [1, 2, 4, 2] is denoted 1 + 2 + 4 + 2, and results in 9, that is, 1 + 2 + 4 + 2 = 9. Because addition is <a href="/facts/Associative/EQX8lV6r">associative</a> and <a href="/facts/Commutative/WaYxJLxd">commutative</a>, 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 <a href="/facts/Function_(mathematics)/CdReEKCh">function</a> 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 <a href="/facts/Natural_number/u65og8JE">natural numbers</a> may be written as 1 + 2 + 3 + 4 + ⋯ + 99 + 100. Otherwise, summation is denoted by using Σ notation, where 
 
 
 
 ∑
 
 
 {\textstyle \sum }
 
 is an enlarged capital <a href="/facts/Greek_letter/1cHwB8Dg">Greek letter</a> <a href="/facts/Sigma/DWcCDzou">sigma</a>. For example, the sum of the first n natural numbers can be denoted as

∑
          
            i
            =
            1
          
          
            n
          
        
        i
      
    
    {\displaystyle \sum _{i=1}^{n}i}

For long summations, and summations of variable length (defined with ellipses or Σ notation), it is a common problem to find <a href="/facts/Closed-form_expression/y0xCXlSk">closed-form expressions</a> for the result. For example,

∑
          
            i
            =
            1
          
          
            n
          
        
        i
        =
        
          
            
              n
              (
              n
              +
              1
              )
            
            2
          
        
        .
      
    
    {\displaystyle \sum _{i=1}^{n}i={\frac {n(n+1)}{2}}.}

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.

Summation open-in-new

Summation