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Home > Series (mathematics)

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

which may or may not be meaningful.

In most cases of interest the terms of the sequence are produced according to a certain rule, e.g., by a formula, by an algorithm, by a sequence of measurements, or even by a random number generator.

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:

1 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 paradoxesZeno's paradoxes are a set of paradoxes conceived by Zeno of Elea to support Parmenides's doctrine that all evidence of the senses is misleading, and particularly that there is no motion. Several of Zeno's eight surviving paradoxes (preserved in Aristotle.

The simplest convergent infinite series is perhaps

It is possible to "visualize" its convergence on the real number lineIn mathematics, the real numbers are intuitively defined as numbers that are in one-to-one correspondence with the points on an infinite line—the number line. The term "real number" is a retronym coined in response to " imaginary number". Real numbers may: 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 complexThe complex numbers are an extension of the real numbers, in which all non-constant polynomials have roots. The complex numbers contain a number , the imaginary unit with , i. is a square root of. Every complex number can be represented in the form , wher) numbers a_n, we say that the series converges towards S or that its value is S if the limitIn mathematics, the concept of a limit is used to describe the behavior of a function, as its argument gets "close" to either some point, or infinity; or the behavior of a sequence's elements, as their index approaches infinity. Limits are used in calculu

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 → ∞.

2 History of the theory of infinite series

2.1 Convergence criteria

The investigation of the validity of infinite series is considered to begin with GaussJohann Carl Friedrich Gauss (Gauss ( April 30, 1777 February 23, 1855) was a legendary German mathematician, astronomer and physicist with a very wide range of contributions; he is considered to be one of the greatest mathematicians of all time. His name. Euler had already considered the hypergeometric seriesIn mathematics, a hypergeometric series could in principle be any formal power series in which the ratio of successive coefficients a ''a is a rational function of n''. In the case of geometric series the ratio is constant. The series for the exponential

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

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 and . 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);

(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.

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Topics: Series (mathematics) Sum Convergence Infinite Converges...

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