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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 synapse biology.

Summation is the addition of a set of numbers; the result is their sum or total. 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.

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.

- $sum\_\{i=m\}^n\; x\_i\; =\; x\_m\; +\; x\_\{m+1\}\; +\; x\_\{m+2\}\; +dots+\; x\_\{n-1\}\; +\; x\_n.$

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

- $sum\_\{k=2\}^6\; k^2\; =\; 2^2+3^2+4^2+5^2+6^2\; =\; 90$.

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

- $sum\; x\_i^2$

- $sum\_\{i=1\}^n\; x\_i^2$.

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:

- $sum\_\{0le\; k<\; 100\}\; f(k)$

- $sum\_\{xin\; S\}\; f(x)$

- $sum\_\{d|n\};mu(d)$

- (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,

- $sum\_\{ell,ell\text{'}\}$

- $sum\_ellsum\_\{ell\text{'}\}.$

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Last updated on Wednesday October 01, 2008 at 18:55:06 PDT (GMT -0700)

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This article is licensed under the GNU Free Documentation License.

Last updated on Wednesday October 01, 2008 at 18:55:06 PDT (GMT -0700)

View this article at Wikipedia.org - Edit this article at Wikipedia.org - Donate to the Wikimedia Foundation

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