en>Christian75: {{advert|date=November 2011}} moved from talk page

2013-02-26T07:43:55Z

{{advert|date=November 2011}} moved from talk page

← Older revision		Revision as of 09:43, 26 February 2013
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	~~I would like to introduce myself to you~~, ~~I am Andrew and my spouse doesn~~'~~t like it at all. Ohio~~ is ~~exactly~~ where ~~his home is~~ and ~~his family members loves it~~. ~~To climb~~ is ~~some thing she would never give up~~. He is ~~an info officer~~.<br><br>~~Review my webpage~~; ~~psychic phone ([http:~~//~~conniecolin~~.~~com~~/xe/~~community~~/~~24580 conniecolin~~.~~com~~])		In the [[complex analysis\|analytic theory]] of [[generalized continued fraction\|continued fractions]], a '''chain sequence''' is an infinite sequence {''a''<sub>''n''</sub>} of non-negative real numbers chained together with another sequence {''g''<sub>''n''</sub>} of non-negative real numbers by the equations

			:<math>
			a_1 = (1-g_0)g_1 \quad a_2 = (1-g_1)g_2 \quad a_n = (1-g_{n-1})g_n
			</math>

			where either (a) 0 ≤ ''g''<sub>''n''</sub> < 1, or (b) 0 < ''g''<sub>''n''</sub> ≤ 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 [[quadratic form\|positive definite]] continued fractions.

			The infinite continued fraction of [[convergence problem#Worpitzky's theorem\|Worpitzky's theorem]] contains a chain sequence. A closely related theorem<ref>[[Hubert Stanley Wall\|Wall]] traces this result back to [[Oskar Perron]] (Wall, 1948, p. 48).</ref> shows that

			:<math>
			f(z) = \cfrac{a_1z}{1 + \cfrac{a_2z}{1 + \cfrac{a_3z}{1 + \cfrac{a_4z}{\ddots}}}} \,
			</math>

			converges uniformly on the closed unit disk \|''z''\| ≤ 1 if the coefficients {''a''<sub>''n''</sub>} are a chain sequence.

			==An example==

			The sequence {¼, ¼, ¼, ...} appears as a limiting case in the statement of Worpitzky's theorem. Since this sequence is generated by setting ''g''<sub>0</sub> = ''g''<sub>1</sub> = ''g''<sub>2</sub> = ...  = ½, it is clearly a chain sequence. This sequence has two important properties.

			*Since ''f''(''x'') = ''x'' − ''x''<sup>2</sup> 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 {''g''<sub>''n''</sub>} = {''x''}, and ''x'' < ½, the resulting sequence {''a''<sub>''n''</sub>} will be an endless repetition of a real number ''y'' that is less than ¼.

			*The choice ''g''<sub>''n''</sub> = ½ is not the only set of generators for this particular chain sequence. Notice that setting

			::<math>
			g_0 = 0 \quad g_1 = {\textstyle\frac{1}{4}} \quad g_2 = {\textstyle\frac{1}{3}} \quad
			g_3 = {\textstyle\frac{3}{8}} \;\dots
			</math>

			:generates the same unending sequence {¼, ¼, ¼, ...}.

			==Notes==
			<references/>

			==References==
			*H. S. Wall, ''Analytic Theory of Continued Fractions'', D. Van Nostrand Company, Inc., 1948; reprinted by Chelsea Publishing Company, (1973), ISBN 0-8284-0207-8

			[[Category:Continued fractions]]

en>DavidCary: link to related articles, etc.

2012-01-21T17:51:32Z

link to related articles, etc.

New page

I would like to introduce myself to you, I am Andrew and my spouse doesn't like it at all. Ohio is exactly where his home is and his family members loves it. To climb is some thing she would never give up. He is an info officer.<br><br>Review my webpage; psychic phone ([http://conniecolin.com/xe/community/24580 conniecolin.com])

T-criterion - Revision history

en>Christian75: {{advert|date=November 2011}} moved from talk page

en>DavidCary: link to related articles, etc.