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mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, summation is the
addition Addition (usually signified by the plus symbol ) is one of the four basic operations of arithmetic, the other three being subtraction, multiplication and division. The addition of two whole numbers results in the total amount or '' sum'' ...
of a
sequence In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is called ...
of any kind of
number A number is a mathematical object used to count, measure, and label. The original examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in language with number words. More universally, individual number ...
s, 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 Matrix most commonly refers to: * ''The Matrix'' (franchise), an American media franchise ** ''The Matrix'', a 1999 science-fiction action film ** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchis ...
,
polynomial In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An exampl ...
s and, in general, elements of any type of
mathematical object A mathematical object is an abstract concept arising in mathematics. In the usual language of mathematics, an ''object'' is anything that has been (or could be) formally defined, and with which one may do deductive reasoning and mathematical p ...
s on which an
operation Operation or Operations may refer to: Arts, entertainment and media * ''Operation'' (game), a battery-operated board game that challenges dexterity * Operation (music), a term used in musical set theory * ''Operations'' (magazine), Multi-Man ...
denoted "+" is defined. Summations of
infinite sequence In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is called t ...
s are called series. They involve the concept of
limit Limit or Limits may refer to: Arts and media * ''Limit'' (manga), a manga by Keiko Suenobu * ''Limit'' (film), a South Korean film * Limit (music), a way to characterize harmony * "Limit" (song), a 2016 single by Luna Sea * "Limits", a 2019 ...
, and are not considered in this article. The summation of an explicit sequence is denoted as a succession of additions. For example, summation of is denoted , and results in 9, that is, . Because addition is
associative In mathematics, the associative property is a property of some binary operations, which means that rearranging the parentheses in an expression will not change the result. In propositional logic, associativity is a valid rule of replacement ...
and
commutative In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. Most familiar as the name of ...
, 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 number In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country"). Numbers used for counting are called '' cardinal ...
s may be written as . Otherwise, summation is denoted by using Σ notation, where \sum is an enlarged capital
Greek letter The Greek alphabet has been used to write the Greek language since the late 9th or early 8th century BCE. It is derived from the earlier Phoenician alphabet, and was the earliest known alphabetic script to have distinct letters for vowels as ...
sigma Sigma (; uppercase Σ, lowercase σ, lowercase in word-final position ς; grc-gre, σίγμα) is the eighteenth letter of the Greek alphabet. In the system of Greek numerals, it has a value of 200. In general mathematics, uppercase Σ is used a ...
. For example, the sum of the first natural numbers can be denoted as \sum_^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 expression In mathematics, a closed-form expression is a mathematical expression that uses a finite number of standard operations. It may contain constants, variables, certain well-known operations (e.g., + − × ÷), and functions (e.g., ''n''th r ...
s for the result. For example, :\sum_^n i = \frac. 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.


Notation


Capital-sigma notation

Mathematical notation uses a symbol that compactly represents summation of many similar terms: the ''summation symbol'', \sum, an enlarged form of the upright capital Greek letter
sigma Sigma (; uppercase Σ, lowercase σ, lowercase in word-final position ς; grc-gre, σίγμα) is the eighteenth letter of the Greek alphabet. In the system of Greek numerals, it has a value of 200. In general mathematics, uppercase Σ is used a ...
. This is defined as :\sum_^n a_i = a_m + a_ + a_ + \cdots + a_ + a_n where is the index of summation; is an indexed variable representing each term of the sum; is the lower bound of summation, and is the upper bound of summation. The "" under the summation symbol means that the index starts out equal to . The index, , is incremented by one for each successive term, stopping when . This is read as "sum of , from to ". Here is an example showing the summation of squares: :\sum_^6 i^2 = 3^2+4^2+5^2+6^2 = 86. In general, while any variable can be used as the index of summation (provided that no ambiguity is incurred), some of the most common ones include letters such as i, j, k, and n; the latter is also often used for the upper bound of a summation. Alternatively, index and bounds of summation are sometimes omitted from the definition of summation if the context is sufficiently clear. This applies particularly when the index runs from 1 to ''n''. For example, one might write that: :\sum a_i^2 = \sum_^n a_i^2. Generalizations of this notation are often used, in which an arbitrary logical condition is supplied, and the sum is intended to be taken over all values satisfying the condition. For example: :\sum_ f(k) is an alternative notation for \sum_^ f(k), the sum of f(k) over all (
integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the languag ...
s) k in the specified range. Similarly, :\sum_ f(x) is the sum of f(x) over all elements x in the set S, and :\sum_\;\mu(d) is the sum of \mu(d) over all positive integers d dividing n. There are also ways to generalize the use of many sigma signs. For example, :\sum_ is the same as :\sum_\sum_. A similar notation is used for the product of a sequence, where \prod, an enlarged form of the Greek capital letter pi, is used instead of \sum.


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 0 (zero) is a number representing an empty quantity. In place-value notation such as the Hindu–Arabic numeral system, 0 also serves as a placeholder numerical digit, which works by Multiplication, multiplying digits to the left of 0 by th ...
, because zero is the
identity Identity may refer to: * Identity document * Identity (philosophy) * Identity (social science) * Identity (mathematics) Arts and entertainment Film and television * ''Identity'' (1987 film), an Iranian film * ''Identity'' (2003 film), an ...
for addition. This is known as the ''
empty sum In mathematics, an empty sum, or nullary sum, is a summation where the number of terms is zero. The natural way to extend non-empty sums is to let the empty sum be the additive identity. Let a_1, a_2, a_3, ... be a sequence of numbers, and let ...
''. These degenerate cases are usually only used when the summation notation gives a degenerate result in a special case. For example, if n=m in the definition above, then there is only one term in the sum; if n=m-1, then there is none.


Formal definition

Summation may be defined recursively as follows: :\sum_^b g(i)=0, for ''b'' < ''a''; : :\sum_^b g(i)=g(b)+\sum_^ g(i), for ''b'' ≥ ''a''.


Measure theory notation

In the notation of measure and integration theory, a sum can be expressed as a
definite integral In mathematics, an integral assigns numbers to functions in a way that describes displacement, area, volume, and other concepts that arise by combining infinitesimal data. The process of finding integrals is called integration. Along with ...
, :\sum_^b f(k) = \int_ f\,d\mu where , b/math> is the
subset In mathematics, set ''A'' is a subset of a set ''B'' if all elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are unequal, then ''A'' is a proper subset of ...
of the integers from a to b, and where \mu is the counting measure.


Calculus of finite differences

Given a function that is defined over the integers in the interval , the following equation holds: :f(n)-f(m)= \sum_^ (f(i+1)-f(i)). This is the analogue of the
fundamental theorem of calculus The fundamental theorem of calculus is a theorem that links the concept of differentiating a function (calculating its slopes, or rate of change at each time) with the concept of integrating a function (calculating the area under its graph, ...
in
calculus of finite differences A finite difference is a mathematical expression of the form . If a finite difference is divided by , one gets a difference quotient. The approximation of derivatives by finite differences plays a central role in finite difference methods for the ...
, which states that: :f(n)-f(m)=\int_m^n f'(x)\,dx, where :f'(x)=\lim_ \frac is the
derivative In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. ...
of . An example of application of the above equation is the following: :n^k=\sum_^ \left((i+1)^k-i^k\right). Using binomial theorem, this may be rewritten as: :n^k=\sum_^ \left(\sum_^ \binom i^j\right). The above formula is more commonly used for inverting of the difference operator \Delta, defined by: :\Delta(f)(n)=f(n+1)-f(n), where is a function defined on the nonnegative integers. Thus, given such a function , the problem is to compute the antidifference of , a function F=\Delta^f such that \Delta F=f. That is, F(n+1)-F(n)=f(n). This function is defined up to the addition of a constant, and may be chosen as''Handbook of Discrete and Combinatorial Mathematics'', Kenneth H. Rosen, John G. Michaels, CRC Press, 1999, . :F(n)=\sum_^ f(i). There is not always a
closed-form expression In mathematics, a closed-form expression is a mathematical expression that uses a finite number of standard operations. It may contain constants, variables, certain well-known operations (e.g., + − × ÷), and functions (e.g., ''n''th r ...
for such a summation, but Faulhaber's formula provides a closed form in the case where f(n)=n^k and, by
linearity Linearity is the property of a mathematical relationship ('' function'') that can be graphically represented as a straight line. Linearity is closely related to '' proportionality''. Examples in physics include rectilinear motion, the linear ...
, for every
polynomial function In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An exa ...
of .


Approximation by definite integrals

Many such approximations can be obtained by the following connection between sums and
integral In mathematics, an integral assigns numbers to functions in a way that describes displacement, area, volume, and other concepts that arise by combining infinitesimal data. The process of finding integrals is called integration. Along with ...
s, which holds for any increasing function ''f'': :\int_^ f(s)\ ds \le \sum_^ f(i) \le \int_^ f(s)\ ds. and for any decreasing function ''f'': :\int_^ f(s)\ ds \le \sum_^ f(i) \le \int_^ f(s)\ ds. For more general approximations, see the Euler–Maclaurin formula. For summations in which the summand is given (or can be interpolated) by an integrable function of the index, the summation can be interpreted as a
Riemann sum In mathematics, a Riemann sum is a certain kind of approximation of an integral by a finite sum. It is named after nineteenth century German mathematician Bernhard Riemann. One very common application is approximating the area of functions or lin ...
occurring in the definition of the corresponding definite integral. One can therefore expect that for instance :\frac\sum_^ f\left(a+i\fracn\right) \approx \int_a^b f(x)\ dx, since the right-hand side is by definition the limit for n\to\infty of the left-hand side. However, for a given summation ''n'' is fixed, and little can be said about the error in the above approximation without additional assumptions about ''f'': it is clear that for wildly oscillating functions the Riemann sum can be arbitrarily far from the Riemann integral.


Identities

The formulae below involve finite sums; for infinite summations or finite summations of expressions involving
trigonometric function In mathematics, the trigonometric functions (also called circular functions, angle functions or goniometric functions) are real functions which relate an angle of a right-angled triangle to ratios of two side lengths. They are widely used in a ...
s or other
transcendental function In mathematics, a transcendental function is an analytic function that does not satisfy a polynomial equation, in contrast to an algebraic function. In other words, a transcendental function "transcends" algebra in that it cannot be expressed ...
s, see list of mathematical series.


General identities

: \sum_^t C\cdot f(n) = C\cdot \sum_^t f(n) \quad (
distributivity In mathematics, the distributive property of binary operations generalizes the distributive law, which asserts that the equality x \cdot (y + z) = x \cdot y + x \cdot z is always true in elementary algebra. For example, in elementary arithmeti ...
) : \sum_^t f(n) \pm \sum_^ g(n) = \sum_^t \left(f(n) \pm g(n)\right)\quad (
commutativity In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. Most familiar as the name of ...
and
associativity In mathematics, the associative property is a property of some binary operations, which means that rearranging the parentheses in an expression will not change the result. In propositional logic, associativity is a valid rule of replacement ...
) : \sum_^t f(n) = \sum_^ f(n-p)\quad (index shift) : \sum_ f(n) = \sum_ f(\sigma(m)), \quad for a
bijection In mathematics, a bijection, also known as a bijective function, one-to-one correspondence, or invertible function, is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other ...
from a finite set onto a set (index change); this generalizes the preceding formula. : \sum_^t f(n) =\sum_^j f(n) + \sum_^t f(n)\quad (splitting a sum, using
associativity In mathematics, the associative property is a property of some binary operations, which means that rearranging the parentheses in an expression will not change the result. In propositional logic, associativity is a valid rule of replacement ...
) : \sum_^f(n)=\sum_^f(n)-\sum_^f(n)\quad (a variant of the preceding formula) : \sum_^t f(n) = \sum_^ f(t-n)\quad (the sum from the first term up to the last is equal to the sum from the last down to the first) : \sum_^t f(n) = \sum_^ f(t-n)\quad (a particular case of the formula above) : \sum_^\sum_^ a_ = \sum_^\sum_^ a_\quad (commutativity and associativity, again) : \sum_ a_ = \sum_^n\sum_^i a_ = \sum_^n\sum_^n a_ = \sum_^\sum_^ a_\quad (another application of commutativity and associativity) : \sum_^ f(n) = \sum_^t f(2n) + \sum_^t f(2n+1)\quad (splitting a sum into its
odd Odd means unpaired, occasional, strange or unusual, or a person who is viewed as eccentric. Odd may also refer to: Acronym * ODD (Text Encoding Initiative) ("One Document Does it all"), an abstracted literate-programming format for describing X ...
and
even Even may refer to: General * Even (given name), a Norwegian male personal name * Even (surname) * Even (people), an ethnic group from Siberia and Russian Far East **Even language, a language spoken by the Evens * Odd and Even, a solitaire game wh ...
parts, for even indexes) : \sum_^ f(n) = \sum_^t f(2n) + \sum_^t f(2n-1)\quad (splitting a sum into its odd and even parts, for odd indexes) :\left(\sum_^ a_i\right) \left(\sum_^ b_j\right)=\sum_^n \sum_^n a_ib_j \quad (
distributivity In mathematics, the distributive property of binary operations generalizes the distributive law, which asserts that the equality x \cdot (y + z) = x \cdot y + x \cdot z is always true in elementary algebra. For example, in elementary arithmeti ...
) : \sum_^m\sum_^n = \left(\sum_^m a_i\right) \left( \sum_^n c_j \right)\quad (distributivity allows factorization) : \sum_^t \log_b f(n) = \log_b \prod_^t f(n)\quad (the
logarithm In mathematics, the logarithm is the inverse function to exponentiation. That means the logarithm of a number  to the base  is the exponent to which must be raised, to produce . For example, since , the ''logarithm base'' 10 ...
of a product is the sum of the logarithms of the factors) : C^ = \prod_^t C^\quad (the
exponential Exponential may refer to any of several mathematical topics related to exponentiation, including: *Exponential function, also: **Matrix exponential, the matrix analogue to the above *Exponential decay, decrease at a rate proportional to value *Expo ...
of a sum is the product of the exponential of the summands)


Powers and logarithm of arithmetic progressions

: \sum_^n c = nc\quad for every that does not depend on : \sum_^n i = \sum_^n i = \frac\qquad (Sum of the simplest
arithmetic progression An arithmetic progression or arithmetic sequence () is a sequence of numbers such that the difference between the consecutive terms is constant. For instance, the sequence 5, 7, 9, 11, 13, 15, . . . is an arithmetic progression with a common differ ...
, consisting of the first ''n'' natural numbers.) : \sum_^n (2i-1) = n^2\qquad (Sum of first odd natural numbers) : \sum_^ 2i = n(n+1)\qquad (Sum of first even natural numbers) : \sum_^ \log i = \log n!\qquad (A sum of
logarithm In mathematics, the logarithm is the inverse function to exponentiation. That means the logarithm of a number  to the base  is the exponent to which must be raised, to produce . For example, since , the ''logarithm base'' 10 ...
s is the logarithm of the product) : \sum_^n i^2 = \sum_^n i^2 = \frac = \frac + \frac + \frac\qquad (Sum of the first squares, see
square pyramidal number In mathematics, a pyramid number, or square pyramidal number, is a natural number that counts the number of stacked spheres in a pyramid with a square base. The study of these numbers goes back to Archimedes and Fibonacci. They are part of a bro ...
.) : \sum_^n i^3 = \left(\sum_^n i \right)^2 = \left(\frac\right)^2 = \frac + \frac + \frac\qquad ( Nicomachus's theorem) More generally, one has Faulhaber's formula for p>1 : \sum_^n k^ = \frac + \fracn^p + \sum_^p \binom p k \frac\,n^, where B_k denotes a
Bernoulli number In mathematics, the Bernoulli numbers are a sequence of rational numbers which occur frequently in analysis. The Bernoulli numbers appear in (and can be defined by) the Taylor series expansions of the tangent and hyperbolic tangent functions, ...
, and \binom p k is a
binomial coefficient In mathematics, the binomial coefficients are the positive integers that occur as coefficients in the binomial theorem. Commonly, a binomial coefficient is indexed by a pair of integers and is written \tbinom. It is the coefficient of the t ...
.


Summation index in exponents

In the following summations, is assumed to be different from 1. : \sum_^ a^i = \frac (sum of a
geometric progression In mathematics, a geometric progression, also known as a geometric sequence, is a sequence of non-zero numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the ''common ratio''. For ex ...
) : \sum_^ \frac = 2-\frac (special case for ) : \sum_^ i a^i =\frac ( times the derivative with respect to of the geometric progression) : \begin \sum_^ \left(b + i d\right) a^i &= b \sum_^ a^i + d \sum_^ i a^i\\ & = b \left(\frac\right) + d \left(\frac\right)\\ & = \frac+\frac \end :::(sum of an arithmetico–geometric sequence)


Binomial coefficients and factorials

There exist very many summation identities involving binomial coefficients (a whole chapter of ''
Concrete Mathematics ''Concrete Mathematics: A Foundation for Computer Science'', by Ronald Graham, Donald Knuth, and Oren Patashnik, first published in 1989, is a textbook that is widely used in computer-science departments as a substantive but light-hearted treatme ...
'' is devoted to just the basic techniques). Some of the most basic ones are the following.


Involving the binomial theorem

: \sum_^n a^ b^i=(a + b)^n, the binomial theorem : \sum_^n = 2^n, the special case where : \sum_^n p^i (1-p)^=1, the special case where , which, for 0 \le p \le 1, expresses the sum of the
binomial distribution In probability theory and statistics, the binomial distribution with parameters ''n'' and ''p'' is the discrete probability distribution of the number of successes in a sequence of ''n'' independent experiments, each asking a yes–no ques ...
: \sum_^ i = n(2^), the value at of the
derivative In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. ...
with respect to of the binomial theorem : \sum_^n \frac = \frac, the value at of the
antiderivative In calculus, an antiderivative, inverse derivative, primitive function, primitive integral or indefinite integral of a function is a differentiable function whose derivative is equal to the original function . This can be stated symbolica ...
with respect to of the binomial theorem


Involving permutation numbers

In the following summations, _P_ is the number of -permutations of . : \sum_^ _P_ = _P_(2^) : \sum_^n _P_ = \sum_^n \prod_^k (i+j) = \frac : \sum_^ i!\cdot = \sum_^ _P_ = \lfloor n! \cdot e \rfloor, \quad n \in \mathbb^+, where and \lfloor x\rfloor denotes the
floor function In mathematics and computer science, the floor function is the function that takes as input a real number , and gives as output the greatest integer less than or equal to , denoted or . Similarly, the ceiling function maps to the least int ...
.


Others

: \sum_^ \binom = \binom : \sum_^ = : \sum_^n i\cdot i! = (n+1)! - 1 : \sum_^n = :\sum_^n ^2 = :\sum_^n \frac = \frac


Harmonic numbers

: \sum_^n \frac = H_n (that is the th
harmonic number In mathematics, the -th harmonic number is the sum of the reciprocals of the first natural numbers: H_n= 1+\frac+\frac+\cdots+\frac =\sum_^n \frac. Starting from , the sequence of harmonic numbers begins: 1, \frac, \frac, \frac, \frac, \do ...
) : \sum_^n \frac = H^k_n (that is a
generalized harmonic number In mathematics, the -th harmonic number is the sum of the reciprocals of the first natural numbers: H_n= 1+\frac+\frac+\cdots+\frac =\sum_^n \frac. Starting from , the sequence of harmonic numbers begins: 1, \frac, \frac, \frac, \frac, \d ...
)


Growth rates

The following are useful
approximation An approximation is anything that is intentionally similar but not exactly equal to something else. Etymology and usage The word ''approximation'' is derived from Latin ''approximatus'', from ''proximus'' meaning ''very near'' and the prefix ' ...
s (using theta notation): : \sum_^n i^c \in \Theta(n^) for real ''c'' greater than −1 : : \sum_^n \frac \in \Theta(\log_e n) (See
Harmonic number In mathematics, the -th harmonic number is the sum of the reciprocals of the first natural numbers: H_n= 1+\frac+\frac+\cdots+\frac =\sum_^n \frac. Starting from , the sequence of harmonic numbers begins: 1, \frac, \frac, \frac, \frac, \do ...
) : : \sum_^n c^i \in \Theta(c^n) for real ''c'' greater than 1 : : \sum_^n \log(i)^c \in \Theta(n \cdot \log(n)^) for
non-negative In mathematics, the sign of a real number is its property of being either positive, negative, or zero. Depending on local conventions, zero may be considered as being neither positive nor negative (having no sign or a unique third sign), or it ...
real ''c'' : : \sum_^n \log(i)^c \cdot i^d \in \Theta(n^ \cdot \log(n)^) for non-negative real ''c'', ''d'' : : \sum_^n \log(i)^c \cdot i^d \cdot b^i \in \Theta (n^d \cdot \log(n)^c \cdot b^n) for non-negative real ''b'' > 1, ''c'', ''d''


History

* In 1675,
Gottfried Wilhelm Leibniz Gottfried Wilhelm (von) Leibniz . ( – 14 November 1716) was a German polymath active as a mathematician, philosopher, scientist and diplomat. He is one of the most prominent figures in both the history of philosophy and the history of ...
, in a letter to
Henry Oldenburg Henry Oldenburg (also Henry Oldenbourg) FRS (c. 1618 as Heinrich Oldenburg – 5 September 1677), was a German theologian, diplomat, and natural philosopher, known as one of the creators of modern scientific peer review. He was one of the fo ...
, suggests the symbol ∫ to mark the sum of differentials (
Latin Latin (, or , ) is a classical language belonging to the Italic languages, Italic branch of the Indo-European languages. Latin was originally a dialect spoken in the lower Tiber area (then known as Latium) around present-day Rome, but through ...
: ''calculus summatorius''), hence the S-shape. The renaming of this symbol to ''
integral In mathematics, an integral assigns numbers to functions in a way that describes displacement, area, volume, and other concepts that arise by combining infinitesimal data. The process of finding integrals is called integration. Along with ...
'' arose later in exchanges with
Johann Bernoulli Johann Bernoulli (also known as Jean or John; – 1 January 1748) was a Swiss mathematician and was one of the many prominent mathematicians in the Bernoulli family. He is known for his contributions to infinitesimal calculus and educating Le ...
. * In 1755, the summation symbol Σ is attested in
Leonhard Euler Leonhard Euler ( , ; 15 April 170718 September 1783) was a Swiss mathematician, physicist, astronomer, geographer, logician and engineer who founded the studies of graph theory and topology and made pioneering and influential discoveries ...
's '' Institutiones calculi differentialis''. Euler uses the symbol in expressions like: : \Sigma \ (2 wx + w^2) = x^2 * In 1772, usage of Σ and Σn is attested by
Lagrange Joseph-Louis Lagrange (born Giuseppe Luigi LagrangiaFourier and
C. G. J. Jacobi Carl Gustav Jacob Jacobi (; ; 10 December 1804 – 18 February 1851) was a German mathematician who made fundamental contributions to elliptic functions, Dynamics (mechanics), dynamics, differential equations, determinants, and number theory. H ...
. Fourier's use includes lower and upper bounds, for example: : \sum_^e^ \ldots


See also

* Capital-pi notation *
Einstein notation In mathematics, especially the usage of linear algebra in Mathematical physics, Einstein notation (also known as the Einstein summation convention or Einstein summation notation) is a notational convention that implies summation over a set of ...
*
Iverson bracket In mathematics, the Iverson bracket, named after Kenneth E. Iverson, is a notation that generalises the Kronecker delta, which is the Iverson bracket of the statement . It maps any statement to a function of the free variables in that statement ...
* Iterated binary operation *
Kahan summation algorithm In numerical analysis, the Kahan summation algorithm, also known as compensated summation, significantly reduces the numerical error in the total obtained by adding a sequence of finite- precision floating-point numbers, compared to the obvious a ...
*
Product (mathematics) In mathematics, a product is the result of multiplication, or an expression that identifies objects (numbers or variables) to be multiplied, called ''factors''. For example, 30 is the product of 6 and 5 (the result of multiplication), and x\cd ...
* Summation by parts * the summation single glyph (U+2211 ''N-ARY SUMMATION'') * the paired glyph's beginning (U+23B2 ''SUMMATION TOP'') * the paired glyph's end (U+23B3 ''SUMMATION BOTTOM'')


Notes


References


Bibliography

*


External links

* {{Authority control Mathematical notation Addition