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Encyclopedia > Summation
For evaluation of sums in closed form see evaluating sums.
Summation is also a term used to describe a process in synapse biology.

Summation is the addition of a set of numbers; the result is their sum. The "numbers" to be summed may be natural numbers, complex numbers, matrices, or still more complicated objects. An infinite sum is a subtle procedure known as a series. Note that the term summation has a special meaning in the context of divergent series related to extrapolation. Sum usually refers to Summation. ... In mathematics, closed form can mean: a finitary expression, rather than one involving (for example) an infinite series, or use of recursion - this meaning usually occurs in a phrase like solution in closed form and one also says closed formula; a closed differential form: see Closed and exact differential forms. ... This article is in need of attention. ... Illustration of the major elements in a prototypical synapse. ... 3 + 2 = 5 with apples, a popular choice in textbooks[1] This article is about addition in mathematics. ... In mathematics, a natural number can mean either an element of the set {1, 2, 3, ...} (i. ... In mathematics, a complex number is a number which is often formally defined to consist of an ordered pair of real numbers , often written: In mathematics, the adjective complex means that the underlying number field is complex numbers, for example complex analysis, complex matrix, complex polynomial and complex Lie algebra. ... In mathematics, a matrix (plural matrices) is a rectangular table of elements (or entries), which may be numbers or, more generally, any abstract quantities that can be added and multiplied. ... In mathematics, a series is often represented as the sum of a sequence of terms. ... 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. ... In mathematics, extrapolation is the process of constructing new data points outside a discrete set of known data points. ...

Contents

Notation

The summation of 1, 2, and 4 is 1 + 2 + 4 = 7. The sum is 7. Since addition is associative, it does not matter whether we interpret "1 + 2 + 4" as (1 + 2) + 4 or as 1 + (2 + 4); the result is the same, so parentheses are usually omitted in a sum. Finite addition is also commutative, so the order in which the numbers are written does not affect its sum. (For issues with infinite summation, see absolute convergence.) In mathematics, associativity is a property that a binary operation can have. ... In mathematics, especially abstract algebra, a binary operation * on a set S is commutative if x * y = y * x for all x and y in S. Otherwise * is noncommutative. ... In mathematics, a series is a sum of a sequence of terms. ...


If a sum has too many terms to be written out individually, the sum may be written with an ellipsis to mark out the missing terms. Thus, the sum of all the natural numbers from 1 to 100 is 1 + 2 + … + 99 + 100 = 5050. This article is about the punctuation symbol. ... Natural number can mean either a positive integer (1, 2, 3, 4, ...) or a non-negative integer (0, 1, 2, 3, 4, ...). Natural numbers have two main purposes: they can be used for counting (there are 3 apples on the table), or they can be used for ordering (this is...


Capital-sigma notation

Mathematical notation has a special representation for compactly representing summation of many similar terms: the summation symbol, a large upright capital Sigma. This is defined thus: Sigma (upper case Σ, lower case σ, alternative ς) is the 18th letter of the Greek alphabet. ...

sum_{i=m}^n x_i = x_m + x_{m+1} + x_{m+2} +cdots+ x_{n-1} + x_n.

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. We could as well have used k instead of i, as in In mathematics, an index is a superscript or subscript to a symbol. ...

sum_{k=2}^6 k^2 = 2^2+3^2+4^2+5^2+6^2 = 90..

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

sum x_i^2

which is informally equivalent to

sum_{i=1}^n x_i^2.

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:

sum_{0le k< 100} f(k)

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

sum_{xin S} f(x)

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

sum_{d|n};mu(d)

is the sum of μ(d) over all integers d dividing n.

(Remark: Although the name of the dummy variable does not matter (by definition), one usually uses letters from the middle of the alphabet (i through q) to denote integers, if there is a risk of confusion. For example, even if there should be no doubt about the interpretation, it could look slightly confusing to many mathematicians to see x instead of k in the above formulae involving k. See also typographical conventions in mathematical formulae.)

There are also ways to generalize the use of many sigma signs. For example, In regression analysis, a dummy variable is one that takes the values 0 or 1 to indicate the absence or presence of some categorical effect that may be expected to shift the outcome. ... Roman letters used in mathematics Greek letters used in mathematics See also: Mathematical alphanumeric symbols Table of mathematical symbols Categories: Mathematics stubs ...

sum_{ell,ell'}

is the same as

sum_ellsum_{ell'}.

Computerized notation

Summations can also be represented in a programming language. A programming language is an artificial language that can be used to control the behavior of a machine, particularly a computer. ...


 sum_{i=m}^{n} x_{i}


is computed by the following Visual Basic/VBScript program: This article is about the Visual Basic language shipping with Microsoft Visual Studio 6. ... VBScript (short for Visual Basic Scripting Edition) is an Active Scripting language developed by Microsoft. ... A computer program is a collection of instructions that describe a task, or set of tasks, to be carried out by a computer. ...

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

or the following C/C++/C#/Java 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: C is a general-purpose, block structured, procedural, imperative computer programming language developed in 1972 by Dennis Ritchie at the Bell Telephone Laboratories for use with the Unix operating system. ... C++ (pronounced see plus plus, IPA: ) is a general-purpose programming language with high-level and low-level capabilities. ... The title given to this article is incorrect due to technical limitations. ... “Java language” redirects here. ...

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

or this Python expression: Python is a high-level programming language first released by Guido van Rossum in 1991. ...

 sum(range(m, n + 1)) 

or this Perl code: Wikibooks has a book on the topic of Perl Programming Perl is a dynamic programming language created by Larry Wall and first released in 1987. ...

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

or the following Fortran (or Matlab) expression: Fortran (previously FORTRAN[1]) is a general-purpose[2], procedural,[3] imperative programming language that is especially suited to numeric computation and scientific computing. ... Not to be confused with Matlab Upazila in Chandpur District, Bangladesh. ...

 sum(x(m:n)) 

And can be rendered in (La)TeX typesetting with:

 sum_{i=m}^n x_i 

Note that most of these examples begin by initializing the sum variable to 0, the identity element for addition. (See "special cases" below). For other uses, see identity (disambiguation). ...


Also note that the traditional sum notation allows for the upper bound to be less than the lower bound. In this case, the index variable is initialized with the upper bound instead of the lower bound, and it is decremented instead of incremented. Since addition is commutative, this might also be accomplished by swapping the upper and lower bound and incrementing in a positive direction as usual.


The exact meaning of sum, and therefore its translation into a programming language, changes depending on the data type of the subscript and upper bound. In other words, sum 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 sum 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.


In the example below:

sum_{xin S} f(x)

x is an iterator, which implies a foreach loop, but S is a set, 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. In mathematics, a set can be thought of as any collection of distinct objects considered as a whole. ...


The return value of sum is a scalar in all examples given above. In computer programming, a return statement causes execution to leave the current subroutine and resume at the point the subroutine was called -- this is called the return address. ... 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. ...


Special cases

It is possible to sum fewer than 2 numbers:

  • If you sum the single term x, then the sum is x.
  • If you sum zero terms, then the 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 + 1, then there is none. For other senses of this word, see zero or 0. ... For other uses, see identity (disambiguation). ... In mathematics, the empty sum, or nullary sum, is the result of adding no numbers. ...


Approximation by definite integrals

Many such approximations can be obtained by the following connection between sums and integrals, which holds for any: This article is about the concept of integrals in calculus. ...


increasing function f: A monotonically increasing function (it is strictly increasing on the left and just non-decreasing on the right). ...

 int_{s=a-1}^{b} f(s), ds le sum_{i=a}^{b} f(i) le int_{s=a}^{b+1} f(s), ds.

decreasing function f: A monotonically increasing function (it is strictly increasing on the left and just non-decreasing on the right). ...

 int_{s=a}^{b+1} f(s), ds le sum_{i=a}^{b} f(i) le int_{s=a-1}^{b} f(s), ds.

For more general approximations, see the Euler-Maclaurin formula. In mathematics, the Euler-Maclaurin formula provides a powerful connection between integrals (see calculus) and sums. ...


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: If you are having difficulty understanding this article, you might wish to learn more about algebra, functions, and mathematical limits. ... In mathematics, a Riemann sum is a method for approximating the values of integrals. ...

 frac{b-a}{n}sum_{i=0}^{n-1} fleft(a+ifrac{b-a}nright)approx int_a^b f(x),dx.

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

 lim_{nrightarrow infty} frac{b-a}{n}sum_{i=0}^{n-1} fleft(a+ifrac{b-a}nright) = int_a^b f(x),dx.

Identities

The following are useful identities:

  • sum_{n=s}^t Csdot f(n) = Csdot sum_{n=s}^t f(n), where 'C' is a distributed constant. (See Scalar multiplication)
  • sum_{n=s}^t f(n) + sum_{n=s}^{t} g(n) = sum_{n=s}^t left[f(n) + g(n)right]
  • sum_{n=s}^t f(n) = sum_{n=s+p}^{t+p} f(n-p)
  • sum_{n=s}^j f(n) + sum_{n=j+1}^t f(n) = sum_{n=s}^t f(n)
  • sum_{i=m}^n x = (n-m+1)x
  • sum_{i=1}^n x = nx, definition of multiplication where n is an integer multiplier to x
  • sum_{i=m}^n i = frac{(n-m+1)(n+m)}{2} (see arithmetic series)
  • sum_{i=0}^n i = sum_{i=1}^n i = frac{(n+1)(n)}{2} (Special case of the arithmetic series)
  • sum_{i=1}^n i^2 = frac{n(n+1)(2n+1)}{6} = frac{n^3}{3} + frac{n^2}{2} + frac{n}{6}
  • sum_{i=1}^n i^3 = left(frac{n(n+1)}{2}right)^2 = frac{n^4}{4} + frac{n^3}{2} + frac{n^2}{4} = left[sum_{i=1}^n iright]^2
  • sum_{i=1}^n i^4 = frac{n(n+1)(2n+1)(3n^2+3n-1)}{30}
  • sum_{i=0}^n i^p = frac{(n+1)^{p+1}}{p+1} + sum_{k=1}^pfrac{B_k}{p-k+1}{pchoose k}(n+1)^{p-k+1}
where Bk is the kth Bernoulli number.

In mathematics, scalar multiplication is one of the basic operations defining a vector space in linear algebra (or more generally, a module in abstract algebra). ... // Definition In mathematics, an arithmetic series is the sum of the components of an arithmetic progression. ... In mathematics, the Bernoulli numbers are a sequence of rational numbers with deep connections in number theory. ... In mathematics, a geometric progression is a sequence of numbers such that the quotient of any two successive members of the sequence is a constant called the common ratio of the sequence. ... In mathematics, particularly in combinatorics, a binomial coefficient is a coefficient of any of the terms in the expansion of the binomial (x+1)n. ... In mathematics, multiplication is an elementary arithmetic operation. ... A limit can be: Limit (mathematics), including: Limit of a function Limit of a sequence Net (topology) Limit (category theory) A constraint (mathematical, physical, economical, legal, etc. ... In mathematics, the binomial theorem is an important formula giving the expansion of powers of sums. ...

Growth rates

The following are useful approximations (using theta notation): It has been suggested that this article or section be merged with estimation. ... For other uses, see Big O. In computational complexity theory, big O notation is often used to describe how the size of the input data affects an algorithms usage of computational resources (usually running time or memory). ...

  • sum_{i=1}^n i^c in Theta(n^{c+1}) for real c greater than -1
  • sum_{i=1}^n frac{1}{i} in Theta(log n)
  • sum_{i=1}^n c^i in Theta(c^n) for real c greater than 1
  • sum_{i=1}^n log(i)^c in Theta(n cdot log(n)^{c}) for nonnegative real c
  • sum_{i=1}^n log(i)^c cdot i^d in Theta(n^{d+1} cdot log(n)^{c}) for nonnegative real c, d
  • sum_{i=1}^n log(i)^c cdot i^d cdot b^i in Theta (n^d cdot log(n)^c cdot b^n) for nonnegative real b > 1, c, d

A negative number is a number that is less than zero, such as &#8722;3. ...

See also

Musical set theory is an atonal or post-tonal method of musical analysis and composition which is based on explaining and proving musical phenomena, taken as sets and subsets, using mathematical rules and notation and using that information to gain insight to compositions or their creation. ... This article or section does not adequately cite its references or sources. ... A checksum is a form of redundancy check, a simple way to protect the integrity of data by detecting errors in data that are sent through space (telecommunications) or time (storage). ... In mathematics, a product is the result of multiplying, or an expression that identifies factors to be multiplied. ...

External links

Derivation of Polynomials Representing the Summation of Natural Numbers Raised to a Positive Whole Number Exponent. http://commons.wikimedia.org/w/index.php?title=Image:Sum_of_i.pdf&oldid=8218130 PlanetMath is a free, collaborative, online mathematics encyclopedia. ...


  Results from FactBites:
 
Summation Notation (0 words)
Often mathematical formulae require the addition of many variables Summation or sigma notation is a convenient and simple form of shorthand used to give a concise expression for a sum of the values of a variable.
A typical element of the sequence which is being summed appears to the right of the summation sign.
The variable of summation is represented by an index which is placed beneath the summation sign.
PlanetMath: summation (1155 words)
It must be noted that, although the running variable usually takes integer values, the summation function needs not, and it can lie on any algebraic structure where a sum is defined.
We now give formulas for evaluating many common summations, which can be combined using the mentioned properties to evaluate a wide range of sums.
This is version 18 of summation, born on 2004-10-12, modified 2007-03-13.
  More results at FactBites »

 
 

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