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]
- Greek: άθροιση
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.
NotationThe 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 notationMathematical 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:
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