Quinhydrone electrode - Revision history

en>2corner at 23:19, 6 December 2014

2014-12-06T23:19:22Z

← Older revision		Revision as of 01:19, 7 December 2014
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	~~In [[mathematics]]~~, ~~an [[infinite geometric series]] of~~ the ~~form~~		Hi there, I am Alyson Boon even though it is not the title on my beginning certification. I've always loved residing in Kentucky but now I'm contemplating other options. Invoicing is what I do for a living but I've always wanted my personal business. To climb is some thing I truly appreciate performing.<br><br>My web site [http://165.132.39.93/xe/visitors/372912 best psychic readings]
	~~:<math>\sum_{k=0}^\infty ar^k = a + ar + ar^2 + ar^3 +\cdots</math>~~
	~~is [[divergent series\|divergent]] if and only if \| ~~''~~r'' \| ≥ [[1 (number)\|1]]~~. ~~Methods~~ for ~~summation of divergent series are sometimes useful, and usually evaluate divergent geometric series to~~ a ~~sum that agrees with the formula for the convergent case~~
	~~:<math>\sum_{k=0}^\infty ar^k = \frac{a}{1-r}</math>~~.
	~~This~~ is true of any summation method that possesses the [[Divergent_series#Properties_of_summation_methods\|properties]] of [[regularity]]{{disambiguation needed\|date=May 2012}}, [[linearity]], and [[Stability theory\|stability]].~~{{Disambiguation needed\|date=August 2011}}~~

	~~==Examples==~~
	~~In increasing order of difficulty to sum:~~
	*[[1 − 1 + 1 − 1 + · · ·]], whose common ratio is [[−1 (number)\|−1]]
	*[[1 − 2 + 4 − 8 + · · ·]], whose common ratio is −2
	*[[1 + 2 + 4 + 8 + · · ·]], whose common ratio is [[2 (number)\|2]]
	*[[1 + 1 + 1 + 1 + · · ·]], whose common ratio is 1.

	~~==Motivation for study==~~
	It is useful to figure out which summation methods produce the geometric series formula for which common ratios. One application for this information is the so-called '''Borel-Okada principle''': If a [[regular summation method]] sums Σ''z''<~~sup~~>~~''n''~~<~~/sup~~> ~~to 1/(1 - ''z'') for all ''z'' in a subset ''S'' of the~~ [~~[complex plane]], given certain restrictions on ''S'', then the method also gives the [[analytic continuation]] of any other function {{nowrap\|1=''f''(''z'') = Σ''a''<sub>''n''<~~/~~sub>''z''<sup>''n''<~~/~~sup>}} on the intersection of ''S'' with the [[Mittag-Leffler star]] for ''f''.<ref>Korevaar p.288</ref>~~

	~~==Summability by region==~~

	~~===Open unit disk===~~
	~~Ordinary summation succeeds only for common ratios \|''z''\| < 1.~~

	~~===Closed unit disk===~~
	*[[Cesàro summation]]
	*[[Abel summation]]

	~~===Larger disks===~~
	*[[Euler summation]]

	~~===Half-plane===~~
	~~The series is [[Borel summation\|Borel summable]] for every ''z'' with real part < 1~~. ~~Any such series is also summable by the generalized Euler method (E, ''a'') for appropriate ''a''~~.

	~~===Shadowed plane===~~
	Certain [[moment constant method]]s besides Borel summation can sum the geometric series on the entire Mittag-Leffler star of the function 1/(1 − ''z''), that is, for all ''z'' except the ray ''z'' ≥ 1.<ref>Moroz p.~~21<~~/~~ref>~~

	~~===Everywhere===~~

	~~==Notes==~~
	~~{{reflist}}~~

	~~==References==~~
	~~<div class="references-small">~~
	*{{cite book \|last=Korevaar \|first=Jacob \|title=Tauberian Theory: A Century of Developments \|publisher=Springer \|year=2004 \|isbn=3-540-21058-X}}
	*{{cite arxiv \|first=Alexander \|last=Moroz \|title=Quantum Field Theory as a Problem of Resummation \|year=1991 \|eprint=hep-th/~~9206074 }}~~
	</~~div>~~

	~~{{Series (mathematics)}}~~

	~~[[Category:Divergent series]]~~
	~~[[Category:Geometric series]~~]

31.205.67.115 at 01:51, 5 December 2013

2013-12-05T01:51:07Z

← Older revision		Revision as of 03:51, 5 December 2013
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	~~Hi there. Let me start by introducing~~ the ~~writer, her name~~ is ~~Sophia~~. ~~For years she's been living in Kentucky but her husband wants them~~ to ~~move~~. ~~Invoicing~~ is ~~my profession~~. ~~To perform lacross~~ is 1 of the ~~things she loves most~~.<br><br>~~Also visit my weblog;~~ [~~http:~~//~~medialab~~.~~zendesk~~.~~com~~/~~entries/54181460~~-~~Will~~-~~You~~-~~Often~~-~~End~~-Up-~~Bored~~-~~Try-One-Of-These-Hobby-Ideas- psychic phone~~]		In [[mathematics]], an [[infinite geometric series]] of the form
			:<math>\sum_{k=0}^\infty ar^k = a + ar + ar^2 + ar^3 +\cdots</math>
			is [[divergent series\|divergent]] if and only if \| ''r'' \| ≥ [[1 (number)\|1]]. Methods for summation of divergent series are sometimes useful, and usually evaluate divergent geometric series to a sum that agrees with the formula for the convergent case
			:<math>\sum_{k=0}^\infty ar^k = \frac{a}{1-r}</math>.
			This is true of any summation method that possesses the [[Divergent_series#Properties_of_summation_methods\|properties]] of [[regularity]]{{disambiguation needed\|date=May 2012}}, [[linearity]], and [[Stability theory\|stability]].{{Disambiguation needed\|date=August 2011}}

			==Examples==
			In increasing order of difficulty to sum:
			*[[1 − 1 + 1 − 1 + · · ·]], whose common ratio is [[−1 (number)\|−1]]
			*[[1 − 2 + 4 − 8 + · · ·]], whose common ratio is −2
			*[[1 + 2 + 4 + 8 + · · ·]], whose common ratio is [[2 (number)\|2]]
			*[[1 + 1 + 1 + 1 + · · ·]], whose common ratio is 1.

			==Motivation for study==
			It is useful to figure out which summation methods produce the geometric series formula for which common ratios. One application for this information is the so-called '''Borel-Okada principle''': If a [[regular summation method]] sums Σ''z''<sup>''n''</sup> to 1/(1 - ''z'') for all ''z'' in a subset ''S'' of the [[complex plane]], given certain restrictions on ''S'', then the method also gives the [[analytic continuation]] of any other function {{nowrap\|1=''f''(''z'') = Σ''a''<sub>''n''</sub>''z''<sup>''n''</sup>}} on the intersection of ''S'' with the [[Mittag-Leffler star]] for ''f''.<ref>Korevaar p.288</ref>

			==Summability by region==

			===Open unit disk===
			Ordinary summation succeeds only for common ratios \|''z''\| < 1.

			===Closed unit disk===
			*[[Cesàro summation]]
			*[[Abel summation]]

			===Larger disks===
			*[[Euler summation]]

			===Half-plane===
			The series is [[Borel summation\|Borel summable]] for every ''z'' with real part < 1. Any such series is also summable by the generalized Euler method (E, ''a'') for appropriate ''a''.

			===Shadowed plane===
			Certain [[moment constant method]]s besides Borel summation can sum the geometric series on the entire Mittag-Leffler star of the function 1/(1 − ''z''), that is, for all ''z'' except the ray ''z'' ≥ 1.<ref>Moroz p.21</ref>

			===Everywhere===

			==Notes==
			{{reflist}}

			==References==
			<div class="references-small">
			*{{cite book \|last=Korevaar \|first=Jacob \|title=Tauberian Theory: A Century of Developments \|publisher=Springer \|year=2004 \|isbn=3-540-21058-X}}
			*{{cite arxiv \|first=Alexander \|last=Moroz \|title=Quantum Field Theory as a Problem of Resummation \|year=1991 \|eprint=hep-th/9206074 }}
			</div>

			{{Series (mathematics)}}

			[[Category:Divergent series]]
			[[Category:Geometric series]]

en>Thehelpfulbot: Robot: Adding missing {{reflist}}

2012-05-10T01:00:18Z

Robot: Adding missing {{reflist}}

New page

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