View source for Limit of a sequence

[[File:Converging Sequence example.svg|320px|thumb|The plot of a convergent sequence {''a<sub>n</sub>''} is shown in blue. Visually we can see that the sequence is converging to the limit 0 as ''n'' increases.]]
[[File:Cauchy sequence illustration.svg|350px|thumb| The plot of a [[Cauchy sequence]] (''x<sub>n</sub>''), shown in blue, as ''x<sub>n</sub>'' versus ''n''. Visually, we see that the sequence appears to be converging to a limit point as the terms in the sequence become closer together as ''n'' increases. In the [[real numbers]] every Cauchy sequence converges to some limit.]]
In [[mathematics]], a [[sequence]] is an ordered list of events.<ref>{{cite web|author= Weisstein, Eric W.|title=Sequence|publisher=MathWorld--A Wolfram Web Resource|url= http://mathworld.wolfram.com/Sequence.html}}</ref> Each event consists of a mathematical object, which is a [[number]] in many cases. In some cases, the sequence tends towards a '''limit'''. In this case, the sequence is said to be '''convergent''', otherwise it is '''divergent'''.<ref>Courant, Richard (1961). ''Differential and Integral Calculus'' Volume I. Glasgow: Blackie & Son, Ltd., p. 29.</ref>

In a convergent sequence, the [[Absolute value|absolute]] difference between the value of the current element and the limit, <math>|x_{n} - lim(X)| </math> will decrease as the sequence progresses. 

==Formal Definition==

We call <math>x</math> the '''limit''' of the sequence <math>(x_n)</math> if the following condition holds:
:*For each [[real number]] <math>\epsilon > 0</math>, there exists a [[natural number]] <math>N</math> such that, for every natural number <math>n > N</math>, we have <math>|x_n - x| < \epsilon</math>. 
In other words, for every measure of closeness <math>\epsilon</math>, the sequence's terms are eventually that close to the limit.  The sequence <math>(x_n)</math> is said to '''converge to''' or '''tend to''' the limit <math>x</math>, written <math>x_n \to x</math> or <math>\lim_{n \to \infty} x_n = x</math>.

If a sequence converges to some limit, then it is '''convergent'''; otherwise it is '''divergent'''.

==History==
The Greek philosopher [[Zeno of Elea]] is famous for formulating [[Zeno's paradoxes|paradoxes that involve limiting processes]].<ref>Craig, Edward (1998) ''Routledge Encyclopedia of Philosophy: Genealogy to Iqbal''. London: Routledge, p. 773.</ref>

[[Leucippus]], [[Democritus]], [[Antiphon (person)|Antiphon]], [[Eudoxus of Cnidus|Eudoxus]] and [[Archimedes]] developed the [[method of exhaustion]], which uses an infinite sequence of approximations to determine an area or a volume. Archimedes succeeded in summing what is now called a [[geometric series]].

[[Isaac Newton|Newton]] dealt with series in his works on ''Analysis with infinite series'' (written in 1669, circulated in manuscript, published in 1711), ''Method of fluxions and infinite series'' (written in 1671, published in English translation in 1736, Latin original published much later) and ''Tractatus de Quadratura Curvarum'' (written in 1693, published in 1704 as an Appendix to his ''Optiks''). In the latter work, Newton considers the binomial expansion of (''x''+''o'')<sup>''n''</sup> which he then linearizes by ''taking limits'' (letting ''o''→0).

In the 18th century, [[mathematician]]s like [[Leonhard Euler|Euler]] succeeded in summing some ''divergent'' series by stopping at the right moment; they did not much care whether a limit existed, as long as it could be calculated. At the end of the century, [[Joseph Louis Lagrange|Lagrange]] in his ''Théorie des fonctions analytiques'' (1797) opined that the lack of rigour precluded further development in calculus. [[Carl Friedrich Gauss|Gauss]] in his etude of [[hypergeometric series]] (1813) for the first time rigorously investigated under which conditions a series converged to a limit.

The modern definition of a limit (for any ε there exists an index ''N'' so that ...) was given by [[Bernhard Bolzano]] (''Der binomische Lehrsatz'', Prague 1816, little noticed at the time) and by [[Karl Weierstraß]] in the 1870s.

==References==
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[[Category:Mathematics]]