Difference between revisions of "Converge"
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| − | In mathematics, a sequence <math> a_n</math> is generally said to '''converge''' to <math>x</math> if roughly speaking as <math> n </math> goes to infinity <math> a_n</math> gets closer and closer to <math>x</math> and stays there. Rigorously, <math> a_n</math> is said to converge to <math>x</math> if for all <math>\epsilon>0</math> there exists <math> N</math> such that for all | + | In mathematics, a sequence <math> a_n</math> is generally said to '''converge''' to <math>x</math> if roughly speaking as <math> n </math> goes to infinity <math> a_n</math> gets closer and closer to <math>x</math> and stays there. Rigorously, <math> a_n</math> is said to converge to <math>x</math> if for all <math>\epsilon>0</math> there exists <math> N</math> such that for all n > N we have <math>|a_n -x| < \epsilon </math>. |
Similar definitions can be made for convergence of functions. | Similar definitions can be made for convergence of functions. | ||
[[Category:Mathematics]] | [[Category:Mathematics]] | ||
Revision as of 00:26, March 14, 2007
In mathematics, a sequence is generally said to converge to if roughly speaking as goes to infinity gets closer and closer to and stays there. Rigorously, is said to converge to if for all there exists Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle N} such that for all n > N we have Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |a_n -x| < \epsilon } .
Similar definitions can be made for convergence of functions.