Editing Limit of a sequence (section)

==History==
The Greek philosopher [[Zeno of Elea]] is famous for formulating [[Zeno's paradoxes|paradoxes that involve limiting processes]].

[[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]].

[[Grégoire de Saint-Vincent]] gave the first definition of limit (terminus) of a [[geometric series]] in his work ''Opus Geometricum'' (1647): "The ''terminus'' of a progression is the end of the series, which none progression can reach, even not if she is continued in infinity, but which she can approach nearer than a given segment."<ref>Van Looy, H. (1984). A chronology and historical analysis of the mathematical manuscripts of Gregorius a Sancto Vincentio (1584–1667). Historia Mathematica, 11(1), 57-75.</ref>

[[Pietro Mengoli]] anticipated the modern idea of limit of a sequence with his study of quasi-proportions in ''Geometriae speciosae elementa'' (1659). He used the term ''quasi-infinite'' for [[Unbounded function|unbounded]] and ''quasi-null'' for [[Vanishing function|vanishing]].

[[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 <math display="inline">(x+o)^n</math>, which he then linearizes by ''taking the limit'' as <math display="inline">o</math> tends to <math display="inline">0</math>.

In the 18th century, [[mathematician]]s such as [[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 study of [[hypergeometric series]] (1813) for the first time rigorously investigated the conditions under which a series converged to a limit.

The modern definition of a limit (for any <math display="inline">\varepsilon</math> there exists an index <math display="inline">N</math> so that ...) was given by [[Bernard Bolzano]] (''Der binomische Lehrsatz'', Prague 1816, which was little noticed at the time), and by [[Karl Weierstrass]] in the 1870s.