Summation operator

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. For example, summation of is denoted , and results in 9, that is, . 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. Summation of a sequence of only one summand results in the summand itself. Summation of an empty sequence (a sequence with no elements), by convention, results in 0.

Very often, the elements of a sequence are defined, through a 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 as . Otherwise, summation is denoted by using Σ notation, where $\sum$ is an enlarged capital Greek letter sigma. For example, the sum of the first natural numbers can be denoted as

:$\sum_^n i$

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. For example,

:$\sum_^n i = \frac.$

Although such formulas do not always exist, many summation formulas have been discovered—with some of the most common and elementary ones being listed in the remainder of this article.

Notation

Capital-sigma notation

Mathematical notation uses a symbol that compactly represents summation of many similar terms: the ''summation symbol'', $\sum$, an enlarged form of the upright capital Greek letter sigma. This is defined as
$\sum_^n a_i = a_m + a_ + a_ + \cdots + a_ + a_n$
where is the "index of summation" or "dummy variable", 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 ". However, some notations may include the index at the upper bound of summation, or omit the indec at the lower bound as in $\sum_ ^ a_i$ or $\sum_m ^n a_i$, respectively. In some cases, there are sigma notation where the range of bounds is omitted, which denotes the dummy variable only, like $\sum_i a_i$. Here is an example showing the summation of squares:
$\sum_^6 i^2 = 3^2+4^2+5^2+6^2 = 86.$
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 $i$, $j$, $k$, and $n$; the latter is also often used for the upper bound of a summation. Alternatively, the 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''. For example, one might write that $\sum a_i = \sum_^n a_i$.

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. For example, $\sum_ f(k)$
is an alternative notation for $\sum_^ f(k),$ the sum of $f(k)$ over all (integers) $k$ in the specified range. Similarly, $\sum_ f(x)$ is the sum of $f(x)$ over all elements $x$ in the set $S$, and $\sum_\;\mu(d)$
is the sum of $\mu(d)$ over all positive integers $d$ dividing $n$.

There are also ways to generalize the use of many sigma notations. For example, one writes double summation as two sigma notations with different dummy variables $\sum_^n \sum_^k a_$. Considering that the both sigma notation's range are the same, the double sigma notations can be wrapped into a single notation, so the double summation is rewritten as $\sum_^n \sum_^n a_ = \sum_^n a_$.

The term is sometimes used when discussing the summation presented above. Contrast to the infinite series, the upper bound tends to infinity $\sum_^\infty a_i$, which results in converge if there is a result of the sum, or diverge if otherwise. The bound in the infinite series's sigma notation can be alternatively denoted as $\sum_ a_i$.

Relatedly, the similar notation is used for the product of a sequence, where $\prod$, an enlarged form of the Greek capital letter pi, is used instead of $\sum$.

Special cases

It is possible to sum fewer than 2 numbers:
* If the summation has one summand $x$, then the evaluated sum is $x$.
* 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 $n=m$ in the definition above, then there is only one term in the sum; if $n=m-1$, then there is none.

Algebraic sum

The phrase 'algebraic sum' refers to a sum of terms which may have positive or negative signs. Terms with positive signs are added, while terms with negative signs are subtracted. e.g.
+1 −1

History

The origin of the summation notation dates back to 1675 when Gottfried Wilhelm Leibniz, in a letter to Henry Oldenburg, suggested the symbol $\int$ to mark the sum of differentials (Latin: ''calculus summatorius''), hence the S-shape. The renaming of this symbol to ''integral'' arose later in exchanges with Johann Bernoulli. In 1755, the summation symbol Σ is attested in Leonhard Euler's '' Institutiones calculi differentialis''. Euler uses the symbol in expressions like $\sum (2wx + w^2) = x^2$. The usage of sigma notation was later attested by mathematicians such as Lagrange

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