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In real analysis, a branch of mathematics, a modulus of convergence is a function that tells how quickly a convergent sequence converges. These moduli are often employed in the study of computable analysis and constructive mathematics. If a sequence of real numbers (''x''''i'') converges to a real number ''x'', then by definition for every real ε > 0 there is a natural number ''N'' such that if ''i'' > ''N'' then |''x'' − ''x''''i''| < ε. A modulus of convergence is essentially a function that, given ε, returns a corresponding value of ''N''. == Definition == Suppose that (''x''''i'') is a convergent sequence of real numbers with limit ''x''. There are two ways of defining a modulus of convergence as a function from natural numbers to natural numbers: * As a function ''f''(''n'') such that for all ''n'', if ''i'' > ''f''(''n'') then |''x'' − ''x''''i''| < 1/''n'' * As a function ''g''(''n'') such that for all ''n'', if ''i'' ≥ ''j'' > ''g''(''n'') then |''x''''i'' − ''x''''j''| < 1/''n'' The latter definition is often employed in constructive settings, where the limit ''x'' may actually be identified with the convergent sequence. Some authors use an alternate definition that replaces 1/''n'' with 2−''n''. 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「modulus of convergence」の詳細全文を読む スポンサード リンク
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