Separation principle - Revision history

131.155.212.127: The given dynamics where wrong I changed it to : $\begin{bmatrix} \dot{x} \\ \dot{e} \\ \end{bmatrix} = \begin{bmatrix} A - B K & BK \\ 0 & A - L C \\ \end{bmatrix} \begin{bmatrix} x \\ e \\ \end{bmatrix}$

2014-08-21T13:07:10Z

The given dynamics where wrong I changed it to : <math>\begin{bmatrix} \dot{x} \\ \dot{e} \\ \end{bmatrix} = \begin{bmatrix} A - B K & BK \\ 0 & A - L C \\ \end{bmatrix} \begin{bmatrix} x \\ e \\ \end{bmatrix}</math>

← Older revision		Revision as of 15:07, 21 August 2014
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89.153.125.222 at 20:20, 9 February 2014

2014-02-09T20:20:19Z

← Older revision		Revision as of 22:20, 9 February 2014
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	[~~[File~~:~~Eye of Horus square~~.~~png\|thumb\|First six summands drawn as portions of a square~~.]]		Nice to satisfy you, my title is Araceli Oquendo but extended car warranty I don't like when people extended [http://www.isoevent.fr/en/node/986034 auto warranty] use my complete title. Her husband and her [http://auto.Howstuffworks.com/buying-selling/cg-extended-warranty-tips.htm selected] to reside in Delaware but she requirements to move simply because of her family members. Bookkeeping is how he supports his family members and his salary has been truly satisfying. What she extended car [http://www.wandmanufaktur.de/index/users/view/id/3122 auto warranty] enjoys performing is bottle tops gathering and she is trying to make it a [http://Www.Carsdirect.com/car-buying/the-best-car-warranty-coverage-plans-top-used-or-new-auto-service-plans profession].<br><br>My blog post ... [http://Bjjoutlet.com/UserProfile/tabid/43/userId/20679/Default.aspx extended auto warranty]

	In [~~[mathematics]], the [[infinite series]] '''1~~/~~2 + 1~~/~~4 + 1~~/~~8 + 1~~/~~16 + · · ·''' is an elementary example~~ of ~~a [[geometric series]] that [[absolute convergence\|converges absolutely]]~~.

	~~Its [[geometric series#Sum\|sum]]~~ is
	~~:<math>\frac12+\frac14+\frac18+\frac{1}{16}+\cdots = \sum_{n=1}^\infty \left({\frac 12}\right)^n = \frac {\frac12}{1-\frac 12} = 1~~. ~~</math>~~

	~~== Direct proof ==~~

	~~As with any~~ [~~[Series (mathematics)\|infinite series]], the infinite sum~~
	:~~<math>\frac12+\frac14+\frac18+\frac{1}{16}+\cdots<~~/~~math>~~
	~~is defined to mean the [[Limit of a sequence\|limit]] of the sum of the first {{mvar\|n}} terms~~
	~~:<math>s_n=\frac12+\frac14+\frac18+\frac{1}{16}+\cdots+\frac{1}{2^n}<~~/~~math>~~
	~~as {{mvar\|n}} approaches infinity~~. ~~Multiplying {{mvar\|s<sub>n<~~/~~sub>}} by 2 reveals a useful relationship:~~
	~~:<math>2s_n = \frac22+\frac24+\frac28+\frac{2}{16}+\cdots+\frac{2}{2^n} = 1+\frac12+\frac14+\frac18+\cdots+\frac{1}{2^{n-1}} = 1+s_n-\frac{1}{2^n}.<~~/~~math>~~
	~~Subtracting {{mvar\|s<sub>n<~~/~~sub>}} from both sides,~~
	~~:<math>s_n = 1-\frac{1}{2^n}.<~~/~~math>~~
	~~As {{mvar\|n}} approaches infinity, {{mvar\|s<sub>n<~~/~~sub>}} [[Limit of a sequence\|tends~~ to~~]] 1.~~

	~~==History==~~
	~~This series was used as~~ a ~~representation of one of [[Zeno's paradoxes]].<ref>~~[http://~~web01~~.~~shu~~.~~edu~~/~~projects~~/~~reals/numser/series.html#zenonpdx Description of Zeno's paradoxes]</ref> The parts of~~ the ~~[[Eye of Horus#In arithmetic\|Eye of Horus~~]~~] were once thought to represent the first six summands of the series~~.<~~ref~~>~~{{cite book \| title=Professor Stewart's Hoard of Mathematical Treasures \| last=Stewart \| first=Ian \| publisher=Profile Books \| ISBN=978 1 84668 292 6 \| year= 2009 \| pages=76–80 }}~~<~~/ref~~>

	~~==See also==~~
	*[[0.~~999~~..~~.]]~~
	*[~~[Dotted note]]~~

	~~==References==~~
	~~{{reflist}}~~

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

	~~{{DEFAULTSORT~~:~~1 2 + 1 4 + 1 8 + 1 16 +}}~~
	~~[[Category:Geometric series]]~~
	~~[[Category:One]]~~


	~~{{mathanalysis-stub}}~~

	~~[[zh:1~~/~~2+1~~/~~4+1~~/~~8+1~~/~~16+…]~~]

193.146.9.193: /* Proof of separation principle for LTI systems */ notation inconsistency and errata on matix

2013-10-01T10:47:46Z

Proof of separation principle for LTI systems: notation inconsistency and errata on matix

← Older revision		Revision as of 12:47, 1 October 2013
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	~~The name~~ of the ~~author~~ is ~~Luther~~. ~~Arizona~~ is ~~her beginning location and she will by no means move~~. ~~Her buddies say it's not good for her but what she enjoys doing~~ is ~~flower arranging and she is attempting~~ to ~~make it~~ a ~~occupation~~. ~~Interviewing is how I make~~ a ~~living and it'~~s ~~some thing I really appreciate~~.<br><br>~~Here is my homepage;~~ [http://~~sun-lan~~.de/~~index~~.~~php?mod~~=~~users&action~~=~~view&id~~=~~14346 sun~~-~~lan.de~~]		[[File:Eye of Horus square.png\|thumb\|First six summands drawn as portions of a square.]]

			In [[mathematics]], the [[infinite series]] '''1/2 + 1/4 + 1/8 + 1/16 + · · ·''' is an elementary example of a [[geometric series]] that [[absolute convergence\|converges absolutely]].

			Its [[geometric series#Sum\|sum]] is
			:<math>\frac12+\frac14+\frac18+\frac{1}{16}+\cdots = \sum_{n=1}^\infty \left({\frac 12}\right)^n = \frac {\frac12}{1-\frac 12} = 1. </math>

			== Direct proof ==

			As with any [[Series (mathematics)\|infinite series]], the infinite sum
			:<math>\frac12+\frac14+\frac18+\frac{1}{16}+\cdots</math>
			is defined to mean the [[Limit of a sequence\|limit]] of the sum of the first {{mvar\|n}} terms
			:<math>s_n=\frac12+\frac14+\frac18+\frac{1}{16}+\cdots+\frac{1}{2^n}</math>
			as {{mvar\|n}} approaches infinity. Multiplying {{mvar\|s<sub>n</sub>}} by 2 reveals a useful relationship:
			:<math>2s_n = \frac22+\frac24+\frac28+\frac{2}{16}+\cdots+\frac{2}{2^n} = 1+\frac12+\frac14+\frac18+\cdots+\frac{1}{2^{n-1}} = 1+s_n-\frac{1}{2^n}.</math>
			Subtracting {{mvar\|s<sub>n</sub>}} from both sides,
			:<math>s_n = 1-\frac{1}{2^n}.</math>
			As {{mvar\|n}} approaches infinity, {{mvar\|s<sub>n</sub>}} [[Limit of a sequence\|tends to]] 1.

			==History==
			This series was used as a representation of one of [[Zeno's paradoxes]].<ref>[http://web01.shu.edu/projects/reals/numser/series.html#zenonpdx Description of Zeno's paradoxes]</ref> The parts of the [[Eye of Horus#In arithmetic\|Eye of Horus]] were once thought to represent the first six summands of the series.<ref>{{cite book \| title=Professor Stewart's Hoard of Mathematical Treasures \| last=Stewart \| first=Ian \| publisher=Profile Books \| ISBN=978 1 84668 292 6 \| year= 2009 \| pages=76–80 }}</ref>

			==See also==
			*[[0.999...]]
			*[[Dotted note]]

			==References==
			{{reflist}}

			{{Series (mathematics)}}

			{{DEFAULTSORT:1 2 + 1 4 + 1 8 + 1 16 +}}
			[[Category:Geometric series]]
			[[Category:One]]


			{{mathanalysis-stub}}

			[[zh:1/2+1/4+1/8+1/16+…]]

en>ZéroBot: r2.7.1) (Robot: Adding pl:Zasada separacji

2012-04-29T00:50:15Z

r2.7.1) (Robot: Adding pl:Zasada separacji

New page

The name of the author is Luther. Arizona is her beginning location and she will by no means move. Her buddies say it's not good for her but what she enjoys doing is flower arranging and she is attempting to make it a occupation. Interviewing is how I make a living and it's some thing I really appreciate.<br><br>Here is my homepage; [http://sun-lan.de/index.php?mod=users&action=view&id=14346 sun-lan.de]

Separation principle - Revision history

131.155.212.127: The given dynamics where wrong I changed it to : \begin{bmatrix} \dot{x} \\ \dot{e} \\ \end{bmatrix} = \begin{bmatrix} A - B K & BK \\ 0 & A - L C \\ \end{bmatrix} \begin{bmatrix} x \\ e \\ \end{bmatrix}

89.153.125.222 at 20:20, 9 February 2014

193.146.9.193: /* Proof of separation principle for LTI systems */ notation inconsistency and errata on matix

en>ZéroBot: r2.7.1) (Robot: Adding pl:Zasada separacji

131.155.212.127: The given dynamics where wrong I changed it to : $\begin{bmatrix} \dot{x} \\ \dot{e} \\ \end{bmatrix} = \begin{bmatrix} A - B K & BK \\ 0 & A - L C \\ \end{bmatrix} \begin{bmatrix} x \\ e \\ \end{bmatrix}$