In mathematics, summation is the addition of a sequence of any kind of numbers, called addends or summands; the result is their sum or total. Beside numbers, other types of values can be summed as well: functions, vectors, matrices, polynomials and, in general, elements of any type of mathematical objects on which an operation denoted "+" is defined. Summations of infinite sequences are called series. They involve the concept of limit, and are not considered in this article. The summation of an explicit sequence is denoted as a succession of additions. For example, summation of is denoted, and results in 9, that is,. Because addition is associative and commutative, there is no need of parentheses, and the result does not depend on the order of the summands. Summation of a sequence of only one element results in this element itself. Summation of an empty sequence results, by convention, in 0. Very often, the elements of a sequence are defined, through regular pattern, as a function of their place in the sequence. For simple patterns, summation of long sequences may be represented with most summands replaced by ellipses. For example, summation of the first 100 natural numbers may be written. Otherwise, summation is denoted by using [|Σ notation], where is an enlarged capital Greek lettersigma. For example, the sum of the first natural integers is denoted For long summations, and summations of variable length, it is a common problem to find closed-form expressions for the result. For example, Although such formulas do not always exist, many summation formulas have been discovered. Some of the most common and elementary ones are listed in this article.
Notation
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 is the index of summation; is an indexed variable representing each term of the sum; is the lower bound of summation, and is the upper bound of summation. The "" under the summation symbol means that the index starts out equal to. The index,, is incremented by one for each successive term, stopping when. This is read as "sum of, from to ". Here is an example showing the summation of squares: 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. Here are some common examples: is the sum of over all 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. The same basic structure is used, with, an enlarged form of the Greek capital letterpi, replacing the.
Special cases
It is possible to sum fewer than 2 numbers:
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.
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 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.
