Sequential Summation Operation

A Sequential Summation Operation is a sequential algebraic operation that involves arithmetic additions.

• AKA: ∑, Sum Of, Sequential Addition.
• Example(s):
  • [math]\displaystyle{ +(1,2,3,4) \equiv +(+(1,4),+(2,3)) \equiv 10 }[/math]
  • [math]\displaystyle{ \sum_{i \mathop =1}^{100}i. }[/math], which btw equals [math]\displaystyle{ (1+100) + (2+99) + ... + (50+51) \equiv 101 \times 50 }[/math]
  • a Weighted Summation Operation.
  • a Sum-of-Squares Operation.
  • …
• Counter-Example(s):
  • Sequential Multiplication Operation.
• See: Algebraic Operation, Partial Sum, Prefix Sum, Running Total, Vector Space, Polynomial, Abelian Group.

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References

2015

• (Wikipedia, 2015) ⇒ http://en.wikipedia.org/wiki/summation Retrieved:2015-1-17.
  • '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).

    The summation of an infinite sequence of values is called a series. A value of such a series may often be defined, by means of a limit (although sometimes the value may be infinite, and often no value results at all). 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 is the sum of each of the members of the sequence. In the example, = 9. Since addition is associative the value does not depend on how the additions are grouped, for instance and 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: . 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:

     :[math]\displaystyle{ \sum_{i \mathop =1}^{100}i. }[/math]

    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

     :[math]\displaystyle{ \sum_{ i \mathop =1}^ni = \frac{n(n+1)}{2} }[/math]

    for all natural numbers n (see Triangular number). 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".

    When it is necessary to clarify that numbers are added with their signs, the term algebraic sum is used. For example, in electric circuit theory Kirchhoff's circuit laws consider the algebraic sum of currents in a network of conductors meeting at a point, assigning opposite signs to currents flowing in and out of the node.