Series (mathematics)

In mathematics, a series is, roughly speaking, an addition of infinitely many terms, one after the other. The study of series is a major part of calculus and its generalization, mathematical analysis. Series are used in most areas of mathematics, even for studying finite structures in combinatorics through generating functions. The mathematical properties of infinite series make them widely applicable in other quantitative disciplines such as physics, computer science, statistics and finance.
Among the Ancient Greeks, the idea that a potentially infinite summation could produce a finite result was considered paradoxical, most famously in Zeno's paradoxes. Nonetheless, infinite series were applied practically by Ancient Greek mathematicians including Archimedes, for instance in the quadrature of the parabola. The mathematical side of Zeno's paradoxes was resolved using the concept of a limit during the 17th century, especially through the early calculus of Isaac Newton. The resolution was made more rigorous and further improved in the 19th century through the work of Carl Friedrich Gauss and Augustin-Louis Cauchy, among others, answering questions about which of these sums exist via the completeness of the real numbers and whether series terms can be rearranged or not without changing their sums using absolute convergence and conditional convergence of series.
In modern terminology, any ordered infinite sequence of terms, whether those terms are numbers, functions, matrices, or anything else that can be added, defines a series, which is the addition of the one after the other. To emphasize that there are an infinite number of terms, series are often also called infinite series to contrast with finite series, a term sometimes used for finite sums. Series are represented by an expression like
or, using capital-sigma summation notation,
The infinite sequence of additions expressed by a series cannot be explicitly performed in sequence in a finite amount of time. However, if the terms and their finite sums belong to a set that has limits, it may be possible to assign a value to a series, called the sum of the series. This value is the limit as tends to infinity of the finite sums of the first terms of the series if the limit exists. These finite sums are called the s of the series. Using summation notation,
if it exists. When the limit exists, the series is convergent or summable and also the sequence is summable, and otherwise, when the limit does not exist, the series is divergent.
The expression denotes both the series—the implicit process of adding the terms one after the other indefinitely—and, if the series is convergent, the sum of the series—the explicit limit of the process. This is a generalization of the similar convention of denoting by both the addition—the process of adding—and its result—the sum of and.
Commonly, the terms of a series come from a ring, often the field of the real numbers or the field of the complex numbers. If so, the set of all series is also itself a ring, one in which the addition consists of adding series terms together term by term and the multiplication is the Cauchy product.

Definition

Series

A series or, redundantly, an infinite series, is an infinite sum. It is often represented as
where the terms are the members of a sequence of numbers, functions, or anything else that can be added. A series may also be represented with capital-sigma notation:
It is also common to express series using a few first terms, an ellipsis, a general term, and then a final ellipsis, the general term being an expression of the th term as a function of :
For example, Euler's number can be defined with the series
where denotes the product of the first positive integers, and is conventionally equal to

Partial sum of a series

Given a series, its th partial sum is
Some authors directly identify a series with its sequence of partial sums. Either the sequence of partial sums or the sequence of terms completely characterizes the series, and the sequence of terms can be recovered from the sequence of partial sums by taking the differences between consecutive elements,
Partial summation of a sequence is an example of a linear sequence transformation, and it is also known as the prefix sum in computer science. The inverse transformation for recovering a sequence from its partial sums is the finite difference, another linear sequence transformation.
Partial sums of series sometimes have simpler closed form expressions, for instance an arithmetic series has partial sums
and a geometric series has partial sums
if or simply if.

Sum of a series

Strictly speaking, a series is said to converge, to be convergent, or to be summable when the sequence of its partial sums has a limit. When the limit of the sequence of partial sums does not exist, the series diverges or is divergent. When the limit of the partial sums exists, it is called the sum of the series or value of the series:
A series with only a finite number of nonzero terms is always convergent. Such series are useful for considering finite sums without taking care of the numbers of terms. When the sum exists, the difference between the sum of a series and its th partial sum, is known as the th truncation error of the infinite series.
An example of a convergent series is the geometric series
It can be shown by algebraic computation that each partial sum is
As one has
the series is convergent and converges to with truncation errors.
By contrast, the geometric series
is divergent in the real numbers. However, it is convergent in the extended real number line, with as its limit and as its truncation error at every step.
When a series's sequence of partial sums is not easily calculated and evaluated for convergence directly, convergence tests can be used to prove that the series converges or diverges.

Grouping and rearranging terms

Grouping

In ordinary finite summations, terms of the summation can be grouped and ungrouped freely without changing the result of the summation as a consequence of the associativity of addition. Similarly, in a series, any finite groupings of terms of the series will not change the limit of the partial sums of the series and thus will not change the sum of the series. However, if an infinite number of groupings is performed in an infinite series, then the partial sums of the grouped series may have a different limit than the original series and different groupings may have different limits from one another; the sum of may not equal the sum of
For example, Grandi's series has a sequence of partial sums that alternates back and forth between and and does not converge. Grouping its elements in pairs creates the series which has partial sums equal to zero at every term and thus sums to zero. Grouping its elements in pairs starting after the first creates the series which has partial sums equal to one for every term and thus sums to one, a different result.
In general, grouping the terms of a series creates a new series with a sequence of partial sums that is a subsequence of the partial sums of the original series. This means that if the original series converges, so does the new series after grouping: all infinite subsequences of a convergent sequence also converge to the same limit. However, if the original series diverges, then the grouped series do not necessarily diverge, as in this example of Grandi's series above. However, divergence of a grouped series does imply the original series must be divergent, since it proves there is a subsequence of the partial sums of the original series which is not convergent, which would be impossible if it were convergent. This reasoning was applied in Oresme's proof of the divergence of the harmonic series, and it is the basis for the general Cauchy condensation test.

Rearrangement

In ordinary finite summations, terms of the summation can be rearranged freely without changing the result of the summation as a consequence of the commutativity of addition. Similarly, in a series, any finite rearrangements of terms of a series does not change the limit of the partial sums of the series and thus does not change the sum of the series: for any finite rearrangement, there will be some term after which the rearrangement did not affect any further terms: any effects of rearrangement can be isolated to the finite summation up to that term, and finite summations do not change under rearrangement.
However, as for grouping, an infinitary rearrangement of terms of a series can sometimes lead to a change in the limit of the partial sums of the series. Series with sequences of partial sums that converge to a value but whose terms could be rearranged to a form a series with partial sums that converge to some other value are called conditionally convergent series. Those that converge to the same value regardless of rearrangement are called unconditionally convergent series.
For series of real numbers and complex numbers, a series is unconditionally convergent if and only if the series summing the absolute values of its terms, is also convergent, a property called absolute convergence. Otherwise, any series of real numbers or complex numbers that converges but does not converge absolutely is conditionally convergent. Any conditionally convergent sum of real numbers can be rearranged to yield any other real number as a limit, or to diverge. These claims are the content of the Riemann series theorem.
A historically important example of conditional convergence is the alternating harmonic series,
which has a sum of the natural logarithm of 2, while the sum of the absolute values of the terms is the harmonic series,
which diverges per the divergence of the harmonic series, so the alternating harmonic series is conditionally convergent. For instance, rearranging the terms of the alternating harmonic series so that each positive term of the original series is followed by two negative terms of the original series rather than just one yields
which is times the original series, so it would have a sum of half of the natural logarithm of 2. By the Riemann series theorem, rearrangements of the alternating harmonic series to yield any other real number are also possible.