In mathematics, summation (capital Greek sigma symbol: ∑) is the addition of a sequence of numbers; the result is their sum or total. If numbers are added sequentially from left to right, any intermediate result is a partial sum, prefix sum, or running total of the summation.
Contents
- Capital sigma notation
- Special cases
- Formal definition
- Measure theory notation
- Fundamental theorem of discrete calculus
- Approximation by definite integrals
- Identities
- General manipulations
- Some summations of polynomial expressions
- Some summations involving exponential terms
- Some summations involving binomial coefficients and factorials
- Growth rates
- References
The numbers to be summed (called addends, or sometimes summands) may be integers, rational numbers, real numbers, or complex numbers. Besides numbers, other types of values can be added as well: vectors, matrices, polynomials and, in general, elements of any additive group (or even monoid).
For finite sequences of such elements, summation always produces a well-defined sum. The summation of an infinite sequence of values is called a series. A value of such a series may often be defined by means of a limit (although sometimes the value may be infinite, and often no value results at all). Another notion involving limits of finite sums is integration.
The summation of the sequence [1, 2, 4, 2] is an expression whose value is the sum of each of the members of the sequence. In the example, 1 + 2 + 4 + 2 = 9. Because addition is associative, the sum does not depend on how the additions are grouped, for instance (1 + 2) + (4 + 2) and 1 + ((2 + 4) + 2) both have the value 9; therefore, parentheses are usually omitted in repeated additions. Addition is also commutative, so permuting the terms of a finite sequence does not change its sum (for infinite summations this property may fail; see Absolute convergence for conditions under which it still holds).
There is no special notation for the summation of such explicit sequences, as the corresponding repeated addition expression will do. There is only a slight difficulty if the sequence has fewer than two elements: the summation of a sequence of one term involves no plus sign (it is indistinguishable from the term itself) and the summation of the empty sequence cannot even be written down (but one can write its value "0" in its place). If, however, the terms of the sequence are given by a regular pattern, possibly of variable length, then a summation operator may be useful or even essential.
For the summation of the sequence of consecutive integers from 1 to 100, one could use an addition expression involving an ellipsis to indicate the missing terms: 1 + 2 + 3 + 4 + ... + 99 + 100. In this case, the reader can easily guess the pattern. However, for more complicated patterns, one needs to be precise about the rule used to find successive terms, which can be achieved by using the summation operator "Σ". Using this sigma notation the above summation is written as:
The value of this summation is 5050. It can be found without performing 99 additions, since it can be shown (for instance by mathematical induction) that
for all natural numbers n. More generally, formulae exist for many summations of terms following a regular pattern.
The term "indefinite summation" refers to the search for an inverse image of a given infinite sequence s of values for the forward difference operator, in other words for a sequence, called antidifference of s, whose finite differences are given by s. By contrast, summation as discussed in this article is called "definite summation".
When it is necessary to clarify that numbers are added with their signs, the term algebraic sum is used. For example, in electric circuit theory Kirchhoff's circuit laws consider the algebraic sum of currents in a network of conductors meeting at a point, assigning opposite signs to currents flowing in and out of the node.
Capital-sigma notation
Mathematical notation uses a symbol that compactly represents summation of many similar terms: the summation symbol, ∑, an enlarged form of the upright capital Greek letter Sigma. This is defined as:
where i represents the index of summation; ai is an indexed variable representing each successive term in the series; m is the lower bound of summation, and n is the upper bound of summation. The "i = m" under the summation symbol means that the index i starts out equal to m. The index, i, is incremented by 1 for each successive term, stopping when i = n.
Here is an example showing the summation of exponential terms (all terms to the power of 2):
Informal writing sometimes omits the definition of the index and bounds of summation when these are clear from context, as in:
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:
is the sum of
is the sum of
is the sum of
There are also ways to generalize the use of many sigma signs. For example,
is the same as
A similar notation is applied when it comes to denoting the product of a sequence, which is similar to its summation, but which uses the multiplication operation instead of addition (and gives 1 for an empty sequence instead of 0). The same basic structure is used, with
Special cases
It is possible to sum fewer than 2 numbers:
These degenerate cases are usually only used when the summation notation gives a degenerate result in a special case. For example, if
Formal definition
Summation may be defined recursively as follows
Measure theory notation
In the notation of measure and integration theory, a sum can be expressed as a definite integral,
where
Fundamental theorem of discrete calculus
Indefinite sums can be used to calculate definite sums with the formula:
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:
decreasing function f:
For more general approximations, see the Euler–Maclaurin formula.
For summations in which the summand is given (or can be interpolated) by an integrable function of the index, the summation can be interpreted as a Riemann sum occurring in the definition of the corresponding definite integral. One can therefore expect that for instance
since the right hand side is by definition the limit for
Identities
The formulae below involve finite sums; for infinite summations or finite summations of expressions involving trigonometric functions or other transcendental functions, see list of mathematical series.
General manipulations
Some summations of polynomial expressions
where
The following formulae are manipulations of
generalized to begin a series at any natural number value (i.e.,
Some summations involving exponential terms
In the summations below a is a constant not equal to 1
Some summations involving binomial coefficients and factorials
There exist very many summation identities involving binomial coefficients (a whole chapter of Concrete Mathematics is devoted to just the basic techniques). Some of the most basic ones are the following.
Growth rates
The following are useful approximations (using theta notation):