# Dictionary Definition

summation

### Noun

1 a concluding summary (as in presenting a case
before a law court) [syn: summing up,
rundown]

2 (physiology) the process whereby multiple
stimuli can produce a response (in a muscle or nerve or other part)
that one stimulus alone does not produce

3 the final aggregate; "the sum of all our
troubles did not equal the misery they suffered" [syn: sum, sum
total]

4 the arithmetic operation of summing;
calculating the sum of two or more numbers; "the summation of four
and three gives seven"; "four plus three equals seven" [syn:
addition, plus]

# User Contributed Dictionary

#### Related terms

#### See also

#### Translations

- Greek: άθροιση

# Extensive Definition

- 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.

## 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

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:- \sum_^n x_i = x_m + x_ + x_ +\cdots+ x_ + 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_^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_^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_

# Synonyms, Antonyms and Related Words

abbreviation, abridgment, abstract, account, accounts, addend, aggregate, amount, apocope, argument, body count, box
score, capitulation, cast, census, compression, condensation, conspectus, core, count, curtailment, difference, elision, ellipsis, epitome, essence, evidence, foreshortening, gist, head count, inventory, main point,
meat, nose count, number, pith, plus, plus sign, precis, product, quantity, recap, recapitulation, reckoning, recount, recounting, reduction, rehearsal, repertory, resume, retrenchment, run-through,
rundown, score, shortening, statement, substance, subtotal, sum, sum and substance, summary, summing, summing up, summing-up,
syncope, synopsis, tale, tally, telescoping, testimony, the bottom line,
the story, the whole story, total, truncation, whole, x number