Summation is the operation of adding a sequence of numbers; the result is their sum or total. If numbers are added sequentially from left to right, any intermediate result is a partial sum, prefix sum, or running total of the summation. The numbers to be summed (called addends, or sometimes summands) may be integers, rational numbers, real numbers, or complex numbers. Besides numbers, other types of values 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 is an expression whose value is the sum of each of the members of the sequence. In the example, 1 + 2 + 4 + 2 = 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 the summation of such explicit sequences, as the corresponding repeated addition expression will do. There is only a slight difficulty if the sequence has fewer than two elements: the summation of a sequence of one term involves no plus sign (it is indistinguishable from the term itself) and the summation of the empty sequence cannot even be written down (but one can write its value "0" in its place). If, however, the terms of the sequence are given by a regular pattern, possibly of variable length, then 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 indicate the missing terms: 1 + 2 + 3 + 4 + ... + 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 sigma notation the above summation is written as:
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
for all natural numbers n. More generally, formulae exist for many summations of terms following a regular pattern.
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".
Other articles related to "sum":
... is practical if and only if and, for every i from 2 to k, where denotes the sum of the divisors of x ... in order to be able to represent as a sum of divisors of n ... of n satisfies the condition above, then any can be represented as a sum of divisors of n, by the following sequence of steps Let, and let ...
... In statistics, the residual sum of squares (RSS) is the sum of squares of residuals ... It is also known as the sum of squared residuals (SSR) or the sum of squared errors of prediction (SSE) ... In general, total sum of squares = explained sum of squares + residual sum of squares ...
... A classic example of Jensen's device is a procedure that computes the sum of a series, real procedure Sum(k, l, u, ak) value l, u integer k, l, u real ak comment k and ak ... compute the first 100 terms of a real array V would be Sum(i, 1, 100, V) ... During the execution of Sum, the actual argument i will increment during each step of the for loop, and each of the procedure's evaluations of ak will ...
... To show the second equality, distribute the product over the sum in the result, every product of primes appears exactly once, so by the fundamental theorem of arithmetic, the sum is equal to ... The sum on the right is the harmonic series, which diverges ...
... is not a collision resistant hash function even if the hash sum file is not tampered with, it is computationally trivial for an attacker to cause deliberate hash ...
Famous quotes containing the word sum:
“the possibility of rule as the sum of rulelessness:”
—Archie Randolph Ammons (b. 1926)
“To sum up:
1. The cosmos is a gigantic fly-wheel making 10,000 revolutions a minute.
2. Man is a sick fly taking a dizzy ride on it.
3. Religion is the theory that the wheel was designed and set spinning to give him the ride.”
—H.L. (Henry Lewis)
“The sum of the whole matter is this, that our civilization cannot survive materially unless it be redeemed spiritually.”
—Woodrow Wilson (18561924)