Sum

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For evaluation of sums in closed form see evaluating sums. Summation is also a term used to describe a process in Chemical synapse#Integration of synaptic inputs.

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, matrix (mathematics), or still more complicated objects. An infinite sum is a subtle procedure known as a series (mathematics). Note that the term summation has a special meaning in the context of divergent series related to extrapolation.

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.)

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.

Capital sigma notation Sums can be represented by the summation symbol, a capital Sigma (letter). This is defined as: \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 (mathematics), i. Here, i represents the index of summation; m is the lower bound of summation, and n is the upper bound of summation.We could as well have used k, as in the following example: \sum_{k=2}^6 k^2 = 2^2+3^2+4^2+5^2+6^2 = 90.

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. 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, \sum_{\ell,\ell'} is the same as \sum_\ell\sum_{\ell'}.

Computerized notation Summations can also be represented in a programming language. \sum_{i=m}^{n} x_{i}is computed by the following Visual Basic/VBScript computer program:Sum = 0 For I = M To N Sum = Sum + X(I) Next I or the following C (programming language)/C++/C_sharp/Java (programming language) 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 {p+1} + \sum_{k=1}^p\frac{B_k}{p-k+1}{p\choose k}(n+1)^{p-k+1} where B_k is the kth Bernoulli number.

\sum_{i=1}^n i^3 = \left i\right^2
\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 x^i = \frac{x}{(1-x)^2} (1-(n+1)x^n+nx^{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 1, c, d