# Series (mathematics)

In mathematics, a**series**is a sum of a sequence of termss. That is, a series is a list of numbers with addition operations between them, e.g,

- 1 + 2 + 3 + 4 + 5 + ...

*infinite*; in the first case they may be handled with elementary algebra, but infinite series require tools from mathematical analysis if they are to be applied in anything more than a tentative way.

Examples of simple series include the arithmetic series which is a sum of an arithmetic progression, written as:

## Infinite series

An **infinite series** is a sum of infinitely many terms. Such a sum can have a finite value; if it has, it is said to *converge*; if it does not, it is said to *diverge*. The fact that infinite series can converge resolves several of Zeno's paradoxes.

The simplest convergent infinite series is perhaps

*equal*to 2 (although it is), but it does prove that it is

*at most*2 — in other words, the series has an upper bound.

This series is a geometric series and mathematicians usually write it as:

*a*

_{n}, we say that the series

**converges towards**or that its

*S***value is**if the limit

*S**S*. If there is no such number, then the series is said to

*diverge*.

Here the sequence of **partial sums** is defined as the sequence

*N*. The definition is the same as saying the sequence of partial sums has limit

*S*, as

*N*→ ∞.

## History of the theory of infinite series

### Convergence criteria

The investigation of the validity of infinite series is considered to begin with Gauss. Euler had already considered the hypergeometric series

Cauchy (1821) insisted on strict tests of convergence; he showed that if two series are convergent their product is not necessarily so, and with him begins the discovery of effective criteria. The terms *convergence* and *divergence* had been introduced long before by Gregory (1668). Euler and Gauss had given various criteria, and Maclaurin had anticipated some of Cauchy's discoveries. Cauchy advanced the theory of power series by his expansion of a complex function in such a form.

Abel (1826) in his memoir on the series

Cauchy's methods led to special rather than general criteria, and the same may be said of Raabe (1832), who made the first elaborate investigation of the subject, of De Morgan (from 1842), whose logarithmic test DuBois-Reymond (1873) and Pringsheim (1889) have shown to fail within a certain region; of Bertrand (1842), Bonnet (1843), Malmsten (1846, 1847, the latter without integration); Stokes (1847), Paucker (1852), Tchebichef (1852), and Arndt (1853). General criteria began with Kummer (1835), and have been studied by Eisenstein (1847), Weierstrass in his various contributions to the theory of functions, Dini (1867), DuBois-Reymond (1873), and many others. Pringsheim's (from 1889) memoirs present the most complete general theory.

### Uniform convergence

The theory of uniform convergence was treated by Cauchy (1821), his limitations being pointed out by Abel, but the first to attack it successfully were Stokes and Seidel (1847-48). Cauchy took up the problem again (1853), acknowledging Abel's criticism, and reaching the same conclusions which Stokes had already found. Thomé used the doctrine (1866), but there was great delay in recognizing the importance of distinguishing between uniform and non-uniform convergence, in spite of the demands of the theory of functions.

### Semi-convergence

Among the early writers was Wronski, whose "loi suprême" (1815) was hardly recognized until Cayley (1873) brought it into prominence.

### Interpolation

### Fourier series

Fourier series were being investigated
as the result of physical considerations at the same time that
Gauss, Abel, and Cauchy were working out the theory of infinite
series. Series for the expansion of sines and cosines, of multiple
arcs in powers of the sine and cosine of the arc had been treated by
Jakob Bernoulli (1702) and his brother Johann (1701) and still
earlier by Viète. Euler and Lagrange had simplified the subject,
as have, more recently, Poinsot, Schröter, Glaisher, and
Kummer. Fourier (1807) set for himself a different problem, to
expand a given function of in terms of the sines or cosines of
multiples of , a problem which he embodied in his *Théorie analytique de la Chaleur* (1822). Euler had already given the
formulas for determining the coefficients in the series; and
Lagrange had passed over them without recognizing their value, but
Fourier was the first to assert and attempt to prove the general
theorem. Poisson (1820-23) also attacked the problem from a
different standpoint. Fourier did not, however, settle the question
of convergence of his series, a matter left for Cauchy (1826) to
attempt and for Dirichlet (1829) to handle in a thoroughly
scientific manner. Dirichlet's treatment (Crelle, 1829), while
bringing the theory of trigonometric series to a temporary
conclusion, has been the subject of criticism and improvement by
Riemann (1854), Heine, Lipschitz, Schläfli, and
DuBois-Reymond. Among other prominent contributors to the theory of
trigonometric and Fourier series have been Dini, Hermite, Halphen,
Krause, Byerly and Appell.

## Some types of infinite series

- A
*geometric series*is one where each successive term is produced by multiplying the previous term by a constant number. Example: 1 + 1/2 + 1/4 + 1/8 + 1/16... - The
*harmonic series*is the series 1 + 1/2 + 1/3 + 1/4 + 1/5... - An
*alternating series*is a series where terms alternate signs. Example: 1 - 1/2 + 1/3 + 1/4 - 1/5...

## Convergence criteria

- If the series ∑
*a*_{n}converges, then the sequence (*a*_{n}) converges to 0 for*n*→∞; the converse is in general not true. - If all the numbers
*a*_{n}are positive and ∑*b*_{n}is a convergent series such that*a*_{n}≤*b*_{n}for all*n*, then ∑*a*_{n}converges as well. If all the*b*_{n}are positive,*a*_{n}≥*b*_{n}for all*n*and ∑*b*_{n}diverges, then ∑*a*_{n}diverges as well. - If the
*a*_{n}are positive and there exists a constant*C*< 1 such that*a*_{n+1}/*a*_{n}≤*C*, then ∑*a*_{n}converges. - If the
*a*_{n}are positive and there exists a constant*C*< 1 such that (*a*_{n})^{1/n}≤*C*, then ∑*a*_{n}converges. - Integral test: if
*f*(*x*) is a positive monotone decreasing function defined on the interval [1, ∞) with*f*(*n*) =*a*_{n}for all*n*, then ∑*a*_{n}converges if and only if the integral ∫_{1}^{∞}*f*(*x*) d*x*is finite. - A series of the form ∑ (-1)
^{n}*a*_{n}(with*a*_{n}≥ 0) is called*alternating*. Such a series converges if the sequence*a*_{n}is monotone decreasing and converges towards 0. The converse is in general not true. - See ratio test.

## Examples

The series

*r*> 1 and diverges for

*r*≤ 1, which can be shown with the integral criterion 5) from above. As a function of

*r*, the sum of this series is Riemann's zeta function.

The geometric series

*z*| < 1.

*b*

_{n}converges to a limit

*L*as

*n*goes to infinity. The value of the series is then

*b*

_{1}-

*L*.

## Absolute convergence

is said to**converge absolutely**if the series of absolute values

If a series converges, but not absolutely, then one can always find a reordering of the terms so that the reordered series diverges. Even more: if the *a*_{n} are real and *S* is any real number, one can find a reordering so that the reordered series converges with limit *S* (Riemann).

## Power series

Several important functions can be represented as Taylor series; these are infinite series involving powers of the independent variable and are also called power series. See also radius of convergence.

Historically, mathematicians such as Leonhard Euler operated liberally with infinite series, even if they were not convergent. When calculus was put on a sound and correct foundation in the nineteenth century, rigorous proofs of the convergence of series were always required. However, the formal operation with non-convergent series has been retained in rings of formal power series which are studied in abstract algebra. Formal power series are also used in combinatorics to describe and study sequences that are otherwise difficult to handle; this is the method of generating functions.

## Generalizations

The notion of series can be defined in every abelian topological group; the most commonly encountered case is that of series in a Banach space.

There is no serious definition for an infinite sum over an uncountable set. For example if *X* is a set and *f* a function on *X* taking non-negative real values, such that

*Y*of

*X*, with

*A*an absolute constant, it follows that

*f*(

*x*) = 0 for all

*x*outside some countable subset of

*X*. In other words, infinite sums of uncountably many non-negative reals make sense only in the case that this is a conventional convergent infinite series, extended by the value 0 to an uncountable set.

Asymptotic series, otherwise asymptotic expansions, are not typically convergent infinite series, but sequences of finite approximations each of which is a good asymptotic representation.