Series (mathematics)

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

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

which may or may not be meaningful. Series may be finite, or 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:

and finite geometric series, a sum of a geometric progression, which can be 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

It is possible to "visualize" its convergence on the real number line: we can imagine a line of length 2, with successive segments marked off of lengths 1, 1/2, 1/4, etc. There is always room to mark the next segment, because the amount of line remaining is always the same as the last segment marked: when we have marked off 1/2, we still have a piece of length 1/2 unmarked, so we can certainly mark the next 1/4. This argument does not prove that the sum is 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:

Formally, if an infinite series

is given with real (or complex) numbers a_n, we say that the series converges towards S or that its value is S if the limit

exists and is equal to 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

indexed by 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

on which Gauss published a memoir in 1812. It established simpler criteria of convergence, and the questions of remainders and the range of convergence.

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

corrected certain of Cauchy's conclusions, and gave a completely scientific summation of the series for complex values of m and x. He showed the necessity of considering the subject of continuity in questions of convergence.

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

Semi-convergent series were studied by Poisson (1823), who also gave a general form for the remainder of the Maclaurin formula. The most important solution of the problem is due, however, to Jacobi (1834), who attacked the question of the remainder from a different standpoint and reached a different formula. This expression was also worked out, and another one given, by Malmsten (1847). Schlömilch (Zeitschrift, Vol.I, p. 192, 1856) also improved Jacobi's remainder, and showed the relation between the remainder and Bernoulli's function  . Genocchi (1852) has further contributed to the theory.

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

Interpolation

Interpolation formulas have been given by various writers from Newton to the present time. Lagrange's theorem is well known, although Euler had already given an analogous form, as are also Olivier's formula (1827), and those of Minding (1830), Cauchy (1837), Jacobi (1845), Grunert (1850, 1853), Christoffel (1858), and Mehler (1864).

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 x in terms of the sines or cosines of multiples of x, 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

1. If the series ∑ a_n converges, then the sequence (a_n) converges to 0 for n→∞; the converse is in general not true.
2. 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.
3. 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.
4. If the a_n are positive and there exists a constant C < 1 such that (a_n)^1/n ≤ C, then ∑ a_n converges.
5. 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) dx is finite.
6. 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.
7. See ratio test.

Examples

The series

converges if 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

converges if and only if |z| < 1.

The telescoping series

converges if the sequence b_n converges to a limit L as n goes to infinity. The value of the series is then b₁ - L.

Absolute convergence

The sum

is said to converge absolutely if the series of absolute values

converges. In this case, the original series, and all reorderings of it, converge, and converge towards the same sum.

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

for any countable subset 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.