Summation

In mathematics, summation is the addition of a sequence 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.

Because addition is associative and commutative, there is no need for parentheses, and the result is the same irrespective of the order of the summands.

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 as 1 + 2 + 3 + 4 + ⋯ + 99 + 100.

is an enlarged capital Greek letter sigma.

For example, the sum of the first n natural numbers can be denoted as For long summations, and summations of variable length (defined with ellipses or Σ notation), it is a common problem to find closed-form expressions for the result.

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.

[1] This is defined as where i is the index of summation; ai is an indexed variable representing each term of the sum; 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 one for each successive term, stopping when i = n.[b] This is read as "sum of ai, from i = m to n".

Here is an example showing the summation of squares: In general, while any variable can be used as the index of summation (provided that no ambiguity is incurred), some of the most common ones include letters such as

Alternatively, index and bounds of summation are sometimes omitted from the definition of summation if the context is sufficiently clear.

This applies particularly when the index runs from 1 to n.[2] For example, one might write that: Generalizations of this notation are often used, in which an arbitrary logical condition is supplied, and the sum is intended to be taken over all values satisfying the condition.

, an enlarged form of the Greek capital letter pi, is used instead of

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.

e.g. +1 -1 Summation may be defined recursively as follows: In the notation of measure and integration theory, a sum can be expressed as a definite integral, where

Given a function f that is defined over the integers in the interval [m, n], the following equation holds: This is known as a telescoping series and is the analogue of the fundamental theorem of calculus in calculus of finite differences, which states that: where is the derivative of f. An example of application of the above equation is the following: Using binomial theorem, this may be rewritten as: The above formula is more commonly used for inverting of the difference operator

This function is defined up to the addition of a constant, and may be chosen as[3] There is not always a closed-form expression for such a summation, but Faulhaber's formula provides a closed form in the case where

and, by linearity, for every polynomial function of n. Many such approximations can be obtained by the following connection between sums and integrals, which holds for any increasing function f: and for any 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

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.

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.

There exist very many summation identities involving binomial coefficients (a whole chapter of Concrete Mathematics is devoted to just the basic techniques).

is the number of k-permutations of n. The following are useful approximations (using theta notation):