Summation is the operation of combining a sequence of numbers using addition Addition is a mathematical operation that represents combining collections of objects together into a larger collection. It is signified by the plus sign . For example, in the picture on the right, there are 3 + 2 apples—meaning three apples and two other apples—which is the same as five apples. Therefore, 3 + 2 = 5. Besides counts of fruit,; the result is their sum or total. An interim An interim is a temporary pause in a line of succession or event. It is frequently used as an appositive noun, in which case it serves as an adjective meaning "in between", "transitional" or "temporary". It may refer to: or present total of a summation process is termed the running total. The numbers to be summed may be integers The integers are formed by the natural numbers including 0 (0, 1, 2, 3, ...) together with the negatives of the non-zero natural numbers (−1, −2, −3, ...). Viewed as a subset of the real numbers, they are numbers that can be written without a fractional or decimal component, and fall within the set {... −2, −1, 0, 1, 2, ...}. For example,, rational numbers In mathematics a rational number is any number that can be expressed as the quotient a/b of two integers, with the denominator b not equal to zero. Since b may be equal to 1, every integer is a rational number. The set of all rational numbers is usually denoted by a boldface Q , which stands for quotient, real numbers In mathematics, the real numbers include both rational numbers, such as 42 and −23/129, and irrational numbers, such as pi and the square root of two; or, a real number can be given by an infinite decimal representation, such as 2.4871773339..., where the digits continue in some way; or, the real numbers may be thought of as points on an, or complex numbers A complex number is a number consisting of a real and imaginary part. It can be written in the form a + bi, where a and b are real numbers, and i is the standard imaginary unit with the property i 2 = −1. The complex numbers contain the ordinary real numbers, but extend them by adding in extra numbers and correspondingly expanding the, and other types of values than numbers can be added as well: vectors A vector space is a mathematical structure formed by a collection of vectors: objects that may be added together and multiplied by numbers, called scalars in this context. Scalars are often taken to be real numbers, but one may also consider vector spaces with scalar multiplication by complex numbers, rational numbers, or even more general fields, matrices An item in a matrix is called an entry or an element. The example has entries 1, 9, 13, 20, 55, and 4. Entries are often denoted by a variable with two subscripts, as shown on the right. Matrices of the same size can be added and subtracted entrywise and matrices of compatible sizes can be multiplied. These operations have many of the properties, polynomials In mathematics, a polynomial is an expression of finite length constructed from variables and constants, using only the operations of addition, subtraction, multiplication, and non-negative, whole-number exponents. For example, x2 − 4x + 7 is a polynomial, but x2 − 4/x + 7x3/2 is not, because its second term involves division by the variable x, 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 In mathematics, the empty sum, or nullary sum, is the result of adding no numbers, in summation for example. Its numerical value is zero).

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 In mathematics, the concept of a "limit" is used to describe the value that a function or sequence "approaches" as the input or index approaches some value. Limits are essential to calculus and are used to define continuity, derivatives and integrals. Such infinite summations are known as series A series is the sum of the terms of a sequence. Finite sequences and series have defined first and last terms, where as infinite sequences and series continue indefinitely. Another notion involving limits of finite sums is integration Integration is an important concept in mathematics and, together with differentiation, is one of the two main operations in calculus. Given a function ƒ of a real variable x and an interval [a, b] of the real line, the definite integral. The term summation has a special meaning related to extrapolation In mathematics, extrapolation is the process of constructing new data points outside a discrete set of known data points. It is similar to the process of interpolation, which constructs new points between known points, but the results of extrapolations are often less meaningful, and are subject to greater uncertainty. It may also mean extension of in the context of divergent series In mathematics, a divergent series is an infinite series that is not convergent, meaning that the infinite sequence of the partial sums of the series does not have a limit.

The summation of the sequence [1, 2, 4, 2] is an expression In mathematics, an expression is a finite combination of symbols that are well-formed according to the rules applicable in the context at hand. Symbols can designate values , variables, operations, relations, or can constitute punctuation or other syntactic entities. The use of expressions can range from simple arithmetic operations like 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 In mathematics, associativity is a property of some binary operations. It means that, within an expression containing two or more occurrences in a row of the same associative operator, the order in which the operations are performed does not matter as long as the sequence of the operands is not changed. That is, rearranging the parentheses in such 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 In mathematics, commutativity is the property that changing the order of something does not change the end result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. The commutativity of simple operations, such as multiplication and addition of numbers, was for many years implicitly assumed and the, so permuting In mathematics, the notion of permutation is used with several slightly different meanings, all related to the act of permuting objects or values. Informally, a permutation of a set of values is an arrangement of those values into a particular order. Thus there are six permutations of the set {1,2,3}, namely [1,2,3], [1,3,2], [2,1,3], [2,3,1], [3,1 the terms of a finite sequence does not change its sum. (For infinite summations this property may fail; see absolute convergence In mathematics, a series is said to converge absolutely if the sum (or integral) of the absolute value of the summand or integrand is finite 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 Ellipsis is a mark or series of marks that usually indicate an intentional omission of a word in the original text. An ellipsis can also be used to indicate a pause in speech, an unfinished thought, or, at the end of a sentence, a trailing off into silence (aposiopesis) (apostrophe and ellipsis mixed). When placed at the end of a sentence, the 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 Mathematical induction is a method of mathematical proof typically used to establish that a given statement is true of all natural numbers. It is done by proving that the first statement in the infinite sequence of statements is true, and then proving that if any one statement in the infinite sequence of statements is true, then so is the next one) that

for all natural numbers In mathematics, there are two conventions for the set of natural numbers: it is either the set of positive integers {1, 2, 3, ...} according to the traditional definition; or the set of non-negative integers{0, 1, 2, ...} according to a definition first appearing in the nineteenth century n. More generally, formulas 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 A finite difference is a mathematical expression of the form f − f(x + a). If a finite difference is divided by b − a, one gets a difference quotient. The approximation of derivatives by finite differences plays a central role in finite difference methods for the numerical solution of differential equations, especially boundary value problems are given by s. By contrast, summation as discussed in this article is called "definite summation".

Notation

Capital-sigma notation

Mathematical notation uses a symbol that compactly represents summation of many similar terms: the summation symbol ∑ (U+2211), an enlarged form of the upright capital Greek letter Sigma Sigma is the eighteenth letter of the Greek alphabet, and carries the 'S' sound. In the system of Greek numerals it has a value of 200. When used at the end of a word, and the word is not all upper case, the final form (ς) is used, e.g. Ὀδυσσεύς (Odysseus) - note the two sigmas in the center of the name, and the word-final sigma at the. This is defined thus:

The subscript gives the symbol for an index variable, i. Here, i represents the index of summation; m is the lower bound of summation, and n is the upper bound of summation. Here i = m under the summation symbol means that the index i starts out equal to m. Successive values of i are found by adding 1 to the previous value of i, stopping when i = n. An example:

.

Informal writing sometimes omits the definition of the index and bounds of summation when these are clear from context, as in

.

One often sees generalizations of this notation in which an arbitrary logical condition is supplied, and the sum is intended to be taken over all values satisfying the condition. For example:

is the sum of f(k) over all (integer) k in the specified range,

is the sum of f(x) over all elements x in the set S, and

is the sum of μ(d) over all positive integers d dividing n.^[1]

There are also ways to generalize the use of many sigma signs. For example,

is the same as

A similar notation is applied when it comes to denoting the product Multiplication is the mathematical operation of scaling one number by another. It is one of the four basic operations in elementary arithmetic (the others being addition, subtraction and division) of a sequence, which is similar to its summation, but which uses the multiplication operation instead of addition (and gives 1 for an empty sequence instead of 0). The same basic structure is used, with ∏, an enlarged form of the Greek capital letter Pi Pi is the sixteenth letter of the Greek alphabet, representing [p]. In the system of Greek numerals it has a value of 80. Letters that arose from pi include Cyrillic Pe (П, п), Coptic pi (Ⲡ, ⲡ), and Gothic pairthra (𐍀), replacing the ∑.

Programming language notation

Summations can also be represented in a programming language A programming language is an artificial language designed to express computations that can be performed by a machine, particularly a computer. Programming languages can be used to create programs that control the behavior of a machine, to express algorithms precisely, or as a mode of human communication. Some languages use a notation for summation similar to the mathematical one. For example, this is Python Python is a general-purpose high-level programming language whose design philosophy emphasizes code readability. Python aims to "[combine] remarkable power with very clear syntax", and its standard library is large and comprehensive. Its use of indentation for block delimiters is unusual among popular programming languages:

sum(x[m:n+1])

and this is the Perl Perl is a high-level, general-purpose, interpreted, dynamic programming language. Perl was originally developed by Larry Wall in 1987 as a general-purpose Unix scripting language to make report processing easier. Since then, it has undergone many changes and revisions and become widely popular amongst programmers. Larry Wall continues to oversee equivalent of the above Python:

use List::Util 'sum';
sum($m..$n);

and this is the PHP PHP: Hypertext Preprocessor is a widely used, general-purpose scripting language that was originally designed for web development to produce dynamic web pages. For this purpose, PHP code is embedded into the HTML source document and interpreted by a web server with a PHP processor module, which generates the web page document. As a general-purpose equivalent of the above Python:

$sum = array_sum($x);

and this is Fortran Fortran is a general-purpose,[note 2] procedural,[note 3] imperative programming language that is especially suited to numeric computation and scientific computing. Originally developed by IBM at their campus in south San Jose, California in the 1950s for scientific and engineering applications, Fortran came to dominate this area of programming (or MATLAB MATLAB is a numerical computing environment and fourth-generation programming language. Developed by MathWorks, MATLAB allows matrix manipulations, plotting of functions and data, implementation of algorithms, creation of user interfaces, and interfacing with programs written in other languages, including C, C++, and Fortran and Octave):

sum(x(m:n))

and this is Haskell Haskell is a standardized, purely functional programming language with non-strict semantics, named after logician Haskell Curry:

sum x

and this is Scheme Scheme is a multi-paradigm programming language. It is one of the two main dialects of Lisp and supports a number of programming paradigms but is best known for its support of functional programming. It was developed by Guy L. Steele and Gerald Jay Sussman in the 1970s. Scheme was introduced to the academic world via a series of papers now:

(apply + x)

and this is Erlang Erlang is a general-purpose concurrent programming language and runtime system. The sequential subset of Erlang is a functional language, with strict evaluation, single assignment, and dynamic typing. For concurrency it follows the Actor model. It was designed by Ericsson to support distributed, fault-tolerant, soft-real-time, non-stop:

lists:sum(X).

and this is J The J programming language, developed in the early 1990s by Ken Iverson and Roger Hui, is a synthesis of APL and the FP and FL function-level languages created by John Backus (or APL APL is an interactive array-oriented language and integrated development environment which is available from a number of commercial and non-commercial vendors and for most computer platforms. It is based on a mathematical notation developed by Kenneth E. Iverson):

+/x

and this is TI-BASIC

sum(seq(x,x,m,n[,1]))
#text in [brackets] is optional

In other languages loops are used, as in the following Visual Basic Visual Basic is the third-generation event-driven programming language and integrated development environment (IDE) from Microsoft for its COM programming model. VB is also considered a relatively easy to learn and use programming language, because of its graphical development features and BASIC heritage/VBScript VBScript is an Active Scripting language developed by Microsoft that is modelled on Visual Basic. It is designed as a “lightweight” language with a fast interpreter for use in a wide variety of Microsoft environments. VBScript uses the Component Object Model to access elements of the environment within which it is running; for example, the program A computer program is a sequence of instructions written to perform a specified task for a computer. A computer requires programs to function, typically executing the program's instructions in a central processor. The program has an executable form that the computer can use directly to execute the instructions. The same program in its human-:

Sum = 0
For I = M To N
Sum = Sum + X(I)
Next I

or the following C C is a general-purpose computer programming language developed in 1972 by Dennis Ritchie at the Bell Telephone Laboratories for use with the Unix operating system/C++/C# C# is a multi-paradigm programming language encompassing imperative, functional, generic, object-oriented (class-based), and component-oriented programming disciplines. It was developed by Microsoft within the .NET initiative and later approved as a standard by Ecma (ECMA-334) and ISO (ISO/IEC 23270). C# is one of the programming languages/Java Java is a programming language originally developed by James Gosling at Sun Microsystems and released in 1995 as a core component of Sun Microsystems' Java platform. The language derives much of its syntax from C and C++ but has a simpler object model and fewer low-level facilities. Java applications are typically compiled to bytecode (class file) code, which assumes that the variables m and n are defined as integer types no wider than int, such that m ≤ n, and that the variable x is defined as an array of values of integer type no wider than int, containing at least m − n + 1 defined elements:

int i;
int sum = 0;
for (i = m; i <= n; i++) {
sum += x[i];
}

In some cases a loop can be written more concisely, as in this Perl code:

$sum += $x[$_] for ($m..$n);

or these alternative Ruby Ruby is a dynamic, reflective, general purpose object-oriented programming language that combines syntax inspired by Perl with Smalltalk-like features. Ruby originated in Japan during the mid-1990s and was initially developed and designed by Yukihiro "Matz" Matsumoto. It is based on Perl, Smalltalk, Eiffel, Ada,[citation needed] and Lisp expressions:

x[m..n].inject{|a,b| a+b}
x[m..n].inject{|a,b| a+b}

or in C++, using its standard library:

std::accumulate(&x[m], &x[n + 1], 0)

when x is a built-in array or a std::vector.

Using LINQ in C# (or other .Net languages), the following code will sum from i to n:

Enumerable.Range(i, n).Aggregate((x , y) => x + y);

Note that most of these examples begin by initializing the sum variable to 0, the identity element In mathematics, an identity element is a special type of element of a set with respect to a binary operation on that set. It leaves other elements unchanged when combined with them. This is used for groups and related concepts for addition. (See "special cases" below).

Also note that the ∑ notation evaluates to a definite value, while most of the loop constructs used above are only valid in an imperative programming language's statement context, requiring the use of an extra variable to hold the final value. It is the variable which would then be used in a larger expression.

The exact meaning of ∑, and therefore its translation into a programming language, changes depending on the data type of the subscript and upper bound. In other words, ∑ is an overloaded symbol In computer science, especially the languages Ada and C++, overloaded expression means that an ambiguous operator expression can only be understood based on the context: see overloading.

In the above examples, the subscript of ∑ was translated into an assignment statement to an index variable at the beginning of a for loop. But the subscript is not always an assignment statement. Sometimes the subscript sets up the iterator for a foreach loop, and sometimes the subscript is itself an array, with no index variable or iterator provided. Other times, the subscript is merely a Boolean expression that contains an embedded variable, implying to a human, but not to a computer, that every value of the variable should be used where the Boolean expression evaluates to true.

In the example below:

x is an iterator, which implies a foreach loop, but S is a set A set is a collection of distinct objects, considered as an object in its own right. Sets are one of the most fundamental concepts in mathematics. Although it was invented at the end of the 19th century, set theory is now a ubiquitous part of mathematics, and can be used as a foundation from which nearly all of mathematics can be derived. In, which is an array-like data structure that can store values of mixed type. The summation routine for a set would have to account for the fact that it is possible to store non-numerical data in a set.

The return value In computer programming, a return statement causes execution to leave the current subroutine and resume at the point in the code immediately after where the subroutine was called — known as its return address. The return address is saved, usually on the process's call stack, as part of the operation of making the subroutine call. Return of ∑ is a scalar In computing, a scalar is a variable or field that can hold only one value at a time; as opposed to composite variables like array, list, record, etc. In some contexts, a scalar value may be understood to be numeric. A scalar data type is the type of a scalar variable. For example, char, int, float, and double are the most common scalar data types in all examples given above.

Special cases

It is possible to sum fewer than 2 numbers:

• If the summation has one summand x, then the evaluated sum is x.
• If the summation has no summands, then the evaluated sum is zero, because zero is the identity for addition. This is known as the empty sum.

These degenerate cases are usually only used when the summation notation gives a degenerate result in a special case. For example, if m = n in the definition above, then there is only one term in the sum; if m > n, then there is none.

Measure theory notation

In the notation of measure & integration theory, a sum can be expressed as a definite integral,

where [a,b] is the subset of the integers from a to b, and where μ is the counting measure.

Fundamental theorem of discrete calculus

Indefinite sums can be used to calculate definite sums with the formula^[2]:

Approximation by definite integrals

Many such approximations can be obtained by the following connection between sums and integrals, which holds for any:

increasing function f:

decreasing function f:

For more general approximations, see the Euler–Maclaurin formula.

For functions that are integrable on the interval [a, b], the Riemann sum can be used as an approximation of the definite integral. For example, the following formula is the left Riemann sum with equal partitioning of the interval:

The accuracy of such an approximation increases with the number n of subintervals, such that:

Identities

The formulas below involve finite sums; for infinite summations see list of mathematical series

General manipulations

, where C is a constant

Some summations of polynomial expressions

(See Harmonic number)

(see arithmetic series)

(Special case of the arithmetic series)

where B_k denotes a Bernoulli number

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