- 1: an indefinite or specified amount of money
- 2: the whole amount : aggregate
- 3: the utmost degree : summit <reached the sum of human happiness>
- 4a : a summary of the chief points or thoughts : summation <the sum of this criticism follows — C. W. Hendel>
- b : gist <the sum and substance of an argument>
- 5a (1) : the result of adding numbers <the sum of 5 and 7 is 12> (2) : the limit of the sum of the first n terms of an infinite series as n increases indefinitely
Summation is the operation of combining a sequence of numbers using addition; the result is their sum or total. An interim or present total of a summation process is termed the running total. The numbers to be summed may be integers, rational numbers, real numbers, or complex numbers, and other types of values than numbers can be added as well: vectors, matrices, polynomials, and in general elements of any additive group (or even monoid). For finite sequences of such elements, summation always produces a well-defined sum (possibly by virtue of the convention for empty sums).
Summation of an infinite sequence of values is not always possible, and when a value can be given for an infinite summation, this involves more than just the addition operation, namely also the notion of a limit. Such infinite summations are known as series. Another notion involving limits of finite sums is integration. The term summation has a special meaning related to extrapolation in the context of divergent series.
The summation of the sequence [1, 2, 4, 2] is an expression whose value, the sum of the sequence, is defined to be that of the repeated addition 1 + 2 + 4 + 2, namely 9. Since addition is associative the value does not depend on how the additions are grouped, for instance (1 + 2) + (4 + 2) and 1 + ((2 + 4) + 2) both have the value 9; therefore, parentheses are usually omitted in repeated additions. Addition is also commutative, so permuting the terms of a finite sequence does not change its sum. (For infinite summations this property may fail; see absolute convergence for conditions under which it still holds.)
There is no special notation for summation of such explicitly given sequences, as the corresponding repeated addition expression will do (but such an expression does not exist for the summation of an empty sequence; one may substitute "0" for such a summation). If, however, the terms of the sequence are given by regular pattern, possibly of variable length, then use of a summation operator may be useful or even essential. For the summation of the sequence of consecutive integers from 1 to 100 one could use an addition expression involving an ellipsis to mark out the missing terms: 1 + 2 + 3 + ... + 99 + 100. In this case the reader easily guesses the pattern; however, for more complicated patterns, one needs to be precise about the rule used to find successive terms, which can be achieved by using the summation operator "Σ". Using this notation the above summation is written
The value of this summation is 5050. It can be found without performing 99 additions, since it can be shown (for instance by mathematical induction) that
The term "indefinite summation" refers to the search for an inverse image of a given infinite sequence s of values for the forward difference operator, in other words for a sequence, called antidifference of s, whose finite differences are given by s. By contrast, summation as discussed in this article is called "definite summation".