Limit of a sequence: Difference between revisions
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imported>Igor Grešovnik m (Started the article) |
imported>Igor Grešovnik m (added definition) |
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The [[Mathematics|mathematical]] concept of '''limit of a sequence''' provides a rigorous definition of the idea of a sequence converging towards a point called the limit. | The [[Mathematics|mathematical]] concept of '''limit of a sequence''' provides a rigorous definition of the idea of a sequence converging towards a point called the limit. | ||
Suppose ''x''<sub>1</sub>, ''x''<sub>2</sub>, ... is a [[sequence]] of [[Real number|real numbers]]. | |||
We say that the real number ''L'' is the ''limit'' of this sequence and we write | |||
:<math> \lim_{n \to \infty} x_n = L </math> | |||
[[if and only if]] for every [[real number]] ε > 0 there exists a [[natural number]] ''n''<sub>0</sub> such that for all ''n'' > ''n''<sub>0</sub> we have |''x''<sub>''n''</sub> − ''L''| < ε. The number ''n''<sub>0</sub> will in general depend on ε. | |||
Revision as of 21:31, 23 November 2007
The mathematical concept of limit of a sequence provides a rigorous definition of the idea of a sequence converging towards a point called the limit.
Suppose x1, x2, ... is a sequence of real numbers. We say that the real number L is the limit of this sequence and we write
if and only if for every real number ε > 0 there exists a natural number n0 such that for all n > n0 we have |xn − L| < ε. The number n0 will in general depend on ε.