summationsəˈmeɪ ʃən
summation (n)
- plural
- summations
English Definitions:
summation, summing up, rundown (noun)
a concluding summary (as in presenting a case before a law court)
summation (noun)
(physiology) the process whereby multiple stimuli can produce a response (in a muscle or nerve or other part) that one stimulus alone does not produce
sum, summation, sum total (noun)
the final aggregate
"the sum of all our troubles did not equal the misery they suffered"
summation, addition, plus (noun)
the arithmetic operation of summing; calculating the sum of two or more numbers
"the summation of four and three gives seven"; "four plus three equals seven"
summation (Noun)
A summarization.
summation (Noun)
An adding up of a series of items.
Summation
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 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. For finite sequences of such elements, summation always produces a well-defined sum. 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 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 + and 1 + 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.
Summation
In mathematics, summation is the addition of a sequence of any kind of numbers, called addends or summands; the result is their sum or total. Beside numbers, other types of values can be summed as well: functions, vectors, matrices, polynomials and, in general, elements of any type of mathematical objects on which an operation denoted "+" is defined. Summations of infinite sequences are called series. They involve the concept of 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 [1, 2, 4, 2] is denoted 1 + 2 + 4 + 2, and results in 9, that is, 1 + 2 + 4 + 2 = 9. Because addition is associative and commutative, there is no need of parentheses, and the result is the same irrespective of the order of the summands. Summation of a sequence of only one element results in this element 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 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 numbers 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 Greek letter sigma. For example, the sum of the first n natural numbers can be denoted as ∑ i = 1 n i . {\textstyle \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 closed-form expressions 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.
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