For evaluation of sums in closed form see evaluating sums. In Mathematics, a series is the sum of the elements of a sequence
Summation is also a term used to describe a process in synapse biology. Chemical synapses are specialized junctions through which Neurons signal to each other and to non-neuronal cells such as those in Muscles or Glands

Summation is the addition of a set of numbers; the result is their sum or total. Addition is the mathematical process of putting things together The "numbers" to be summed may be natural numbers, complex numbers, matrices, or still more complicated objects. In Mathematics, a natural number (also called counting number) can mean either an element of the set (the positive Integers or an Complex plane In Mathematics, the complex numbers are an extension of the Real numbers obtained by adjoining an Imaginary unit, denoted In Mathematics, a matrix (plural matrices) is a rectangular table of elements (or entries) which may be Numbers or more generally An infinite sum is a subtle procedure known as a series. In Mathematics, a series is often represented as the sum of a Sequence of terms That is a series is represented as a list of numbers with Note that the term summation has a special meaning in the context of divergent series related to extrapolation. In Mathematics, a divergent series is an Infinite series that is not convergent, meaning that the infinite Sequence of the Partial sums In Mathematics, extrapolation is the process of constructing new data points outside a Discrete set of known data points

## 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. In Mathematics, associativity is a property that a Binary operation can have Finite addition is also commutative, so the order in which the numbers are written does not affect its sum. In Mathematics, commutativity is the ability to change the order of something without changing the end result (For issues with infinite summation, see absolute convergence. In Mathematics, a series (or sometimes also an Integral) is said to converge absolutely if the sum (or integral of the Absolute value of the )

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. Ellipsis (plural ellipses; from Greek 'omission' in Printing and Writing refers to a mark or series of marks that usually indicate an intentional Thus, the sum of all the natural numbers from 1 to 100 is 1 + 2 + … + 99 + 100 = 5050. In Mathematics, a natural number (also called counting number) can mean either an element of the set (the positive Integers or an

### 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. Sigma (upper case Σ, lower case σ; Greek Σιγμα lower case in word-final position ς) is the eighteenth letter of the Greek This is defined thus:

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

The subscript gives the symbol for an index variable, i. The word index is used in variety of senses in Mathematics. In perhaps the most frequent sense an index is a Superscript 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

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

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:

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

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

$\sum_{x\in S} f(x)$

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. In Regression analysis, a dummy variable (also known as indicator variable or just dummy) is one that takes the values 0 or 1 to indicate the absence 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. Typographical conventions in mathematical formulae provide uniformity across mathematical texts and help the readers of those texts to grasp new concepts quickly )

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

$\sum_{\ell,\ell'}$

is the same as

$\sum_\ell\sum_{\ell'}.$

### Programming language notation

Summations can also be represented in a programming language. A programming language is an Artificial language that can be used to write programs which control the behavior of a machine particularly a Computer. $\sum_{i=m}^{n} x_{i}$ Some languages use a notation for summation similar to the mathematical one. For example, this is Python:

sum(x[m:n+1])

and this is Fortran (or Matlab):

sum(x(m:n))

In other languages you have to use loops, as in the following Visual Basic/VBScript program:

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:

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:

$sum = 0;$sum += $x[$_] for ($m. Python is a general-purpose High-level programming language. Its design philosophy emphasizes programmer productivity and code readability Fortran (previously FORTRAN) is a general-purpose, procedural, imperative Programming language that is especially suited to MATLAB is a numerical computing environment and Programming language. Visual Basic ( VB) is the third-generation event-driven programming language and associated development environment (IDE from VBScript (short for Visual Basic Scripting Edition) is an Active Scripting language developed by Microsoft. Computer programs (also software programs, or just programs) are instructions for a Computer. tags please moot on the talk page first! --> In Computing, C is a general-purpose cross-platform block structured C++ (" C Plus Plus " ˌsiːˌplʌsˈplʌs is a general-purpose Programming language. C# (pronounced C Sharp is a Multi-paradigm NOTES FOR EDITORS "Perl" is not an acronym (read the "Name" section below .$n);

or these alternative Ruby expressions:

x[m. Ruby is a dynamic, reflective, general purpose Object-oriented programming language that combines syntax inspired by Perl with Smalltalk . n]. inject{|a,b| a+b}x[m. . n]. inject(0){|a,b| a+b}

or in C++, using its standard library:

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

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

Note that most of these examples begin by initializing the sum variable to 0, the identity element for addition. In Mathematics, an identity element (or neutral element) is a special type of element of a set with respect to a Binary operation on that (See "special cases" below).

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

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:

$\sum_{x\in S} f(x)$

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.

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 in the code immediately after where the subroutine 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

### 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. In Mathematics, an identity element (or neutral element) is a special type of element of a set with respect to a Binary operation on that This is known as the empty sum. In Mathematics, the empty sum, or nullary sum, is the result of adding no numbers in Summation for example

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.

## 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:

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

decreasing function f:

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

For more general approximations, see the Euler–Maclaurin formula. The European Space Agency 's INTErnational Gamma-Ray Astrophysics Laboratory ( INTEGRAL) is detecting some of the most energetic radiation that comes from space 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. In the branch of Mathematics known as Real analysis, the Riemann integral, created by Bernhard Riemann, was the first rigorous definition of the Integral In Mathematics, a Riemann sum is a method for approximating the total area underneath a curve on a graph otherwise known as an Integral. For example, the following formula is the left Riemann sum with equal partitioning of the interval:

$\frac{b-a}{n}\sum_{i=0}^{n-1} f\left(a+i\frac{b-a}n\right)\approx \int_a^b f(x)\,dx.$

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

$\lim_{n\rightarrow \infty} \frac{b-a}{n}\sum_{i=0}^{n-1} f\left(a+i\frac{b-a}n\right) = \int_a^b f(x)\,dx.$

## Identities

The following are useful identities:

• $\sum_{n=s}^t C\sdot f(n) = C\sdot \sum_{n=s}^t f(n)$, 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)
• $\sum_{i=1}^n i^2 = \frac{n(n+1)(2n+1)}{6} = \frac{n^3}{3} + \frac{n^2}{2} + \frac{n}{6}$
• $\sum_{i=1}^n i^3 = \left(\frac{n(n+1)}{2}\right)^2 = \frac{n^4}{4} + \frac{n^3}{2} + \frac{n^2}{4} = \left[\sum_{i=1}^n i\right]^2$
• $\sum_{i=1}^n i^4 = \frac{n(n+1)(2n+1)(3n^2+3n-1)}{30} = \frac{n^5}{5} + \frac{n^4}{2} + \frac{n^3}{3} - \frac{n}{30}$
• $\sum_{i=0}^n i^p = \frac{(n+1)^{p+1}}{p+1} + \sum_{k=1}^p\frac{B_k}{p-k+1}{p\choose k}(n+1)^{p-k+1}$
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 In Mathematics, an arithmetic progression or arithmetic sequence is a Sequence of Numbers such that the difference of any two successive members In Mathematics, the Bernoulli numbers are a Sequence of Rational numbers with deep connections to Number theory.
• $\sum_{i=m}^n x^i = \frac{x^{n+1}-x^{m}}{x-1}$ (see geometric series)
• $\sum_{i=0}^n x^i = \frac{x^{n+1}-1}{x-1}$ (special case of the above where m = 0)
• $\sum_{i=0}^n i 2^i = 2+2^{n+1}(n-1)$
• $\sum_{i=0}^n i^2 x^i = \frac{x}{(1-x)^3} (1+x-(n+1)^2x^n+(2n^2+2n-1)x^{n+1}-n^2x^{n+2})$
• $\sum_{i=0}^n {n \choose i} = 2^n$ (see binomial coefficient)
• $\sum_{i=0}^{n-1} {i \choose k} = {n \choose k+1}$
• $\left(\sum_i a_i\right)\left(\sum_j b_j\right) = \sum_i\sum_j a_ib_j$
• $\left(\sum_i a_i\right)^2 = 2\sum_i\sum_{j
• $\sum_{n=a}^b f(n) = \sum_{n=b}^{a} f(n)$
• $\sum_{n=s}^t f(n) = \sum_{n=-t}^{-s} f(-n)$
• $\sum_{n=0}^t f(2n) + \sum_{n=0}^t f(2n+1) = \sum_{n=0}^{2t+1} f(n)$
• $\sum_{n=0}^t \sum_{i=0}^{z-1} f(z\sdot n+i) = \sum_{n=0}^{z\sdot t+z-1} f(n)$
• $\widehat{b}^{\left[\sum_{n=s}^t f(n) \right]} = \prod_{n=s}^t \widehat{b}^{f(n)}$ (See Product of a series)
• $\lim_{t\rightarrow \infty} \sum_{n=a}^t f(n) = \sum_{n=a}^\infty f(n)$ (See Infinite limits)
• $(a + b)^n = \sum_{i=0}^n {n \choose i}a^{(n-i)} b^i$, for binomial expansion
• $\sum_{n=b+1}^{\infty} \frac{b}{n^2 - b^2} = \sum_{n=1}^{2b} \frac{1}{2n}$

## Growth rates

The following are useful approximations (using theta notation):

• $\sum_{i=1}^n i^c \in \Theta(n^{c+1})$ 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
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