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As the positive integer ''n'' becomes larger and larger, the value ''n'' sin(1/''n'') becomes arbitrarily close to 1. We say that "the limit of the sequence ''n'' sin(1/''n'') equals 1." In mathematics, the limit of a sequence is the value that the terms of a sequence "tend to".〔Courant (1961), p. 29.〕 If such a limit exists, the sequence is called convergent. A sequence which does not converge is said to be divergent.〔Courant (1961), p. 39.〕 The limit of a sequence is said to be the fundamental notion on which the whole of analysis ultimately rests.〔 Limits can be defined in any metric or topological space, but are usually first encountered in the real numbers. ==History== The Greek philosopher Zeno of Elea is famous for formulating paradoxes that involve limiting processes. Leucippus, Democritus, Antiphon, 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. 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'')''n'' which he then linearizes by ''taking limits'' (letting ''o''→0). In the 18th century, mathematicians such as 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, Lagrange in his ''Théorie des fonctions analytiques'' (1797) opined that the lack of rigour precluded further development in calculus. 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. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「limit of a sequence」の詳細全文を読む スポンサード リンク
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