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In the analytic theory of continued fractions, a chain sequence is an infinite sequence of non-negative real numbers chained together with another sequence of non-negative real numbers by the equations : where either (a) 0 ≤ ''g''''n'' < 1, or (b) 0 < ''g''''n'' ≤ 1. Chain sequences arise in the study of the convergence problem – both in connection with the parabola theorem, and also as part of the theory of positive definite continued fractions. The infinite continued fraction of Worpitzky's theorem contains a chain sequence. A closely related theorem〔Wall traces this result back to Oskar Perron (Wall, 1948, p. 48).〕 shows that : converges uniformly on the closed unit disk |''z''| ≤ 1 if the coefficients are a chain sequence. ==An example== The sequence appears as a limiting case in the statement of Worpitzky's theorem. Since this sequence is generated by setting ''g''0 = ''g''1 = ''g''2 = ... = ½, it is clearly a chain sequence. This sequence has two important properties. *Since ''f''(''x'') = ''x'' − ''x''2 is a maximum when ''x'' = ½, this example is the "biggest" chain sequence that can be generated with a single generating element; or, more precisely, if = , and ''x'' < ½, the resulting sequence will be an endless repetition of a real number ''y'' that is less than ¼. *The choice ''g''''n'' = ½ is not the only set of generators for this particular chain sequence. Notice that setting :: :generates the same unending sequence . 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「Chain sequence」の詳細全文を読む スポンサード リンク
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