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 term summation has a special meaning related to extrapolation in the context of divergent series.
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 easily guesses 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 numbersn. More generally, formulae exist for many summations of terms following a regular pattern.
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.
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 over all (integers) in the specified range,
is the sum of over all elements in the set , and
is the sum of over all positive integers dividing .
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 , an enlarged form of the Greek capital letter Pi, replacing the .
If the summation has one summand , then the evaluated sum is .
If the summation has no summands, then the evaluated sum is zero, because zero is the identity for addition. This is known as the empty sum.
These degenerate cases are usually only used when the summation notation gives a degenerate result in a special case. For example, if in the definition above, then there is only one term in the sum; if , then there is none.
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 of the left hand side. However, for a given summation n is fixed, and little can be said about the error in the above approximation without additional assumptions about f: it is clear that for wildly oscillating functions the Riemann sum can be arbitrarily far from the Riemann integral.
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.
^Although the name of the dummy variable does not matter (by definition), one usually uses letters from the middle of the alphabet ( through ) 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 instead of in the above formulae involving . See also typographical conventions in mathematical formulae.
^"Handbook of discrete and combinatorial mathematics", Kenneth H. Rosen, John G. Michaels, CRC Press, 1999, ISBN 0-8493-0149-1