Weierstrass function - Revision history

en>Mhauwe: /* Construction */ properly define f

2014-11-27T21:14:22Z

Construction: properly define f

← Older revision		Revision as of 23:14, 27 November 2014
Line 1:		Line 1:
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en>Mshrieve: /* Construction */

2014-02-11T04:27:50Z

Construction

← Older revision		Revision as of 06:27, 11 February 2014
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	In [~~[mathematics]~~], the ~~'''Weierstrass M-test''' is a test for showing that an [[infinite series]]~~ of ~~[[function (mathematics)\|functions]] converges [[uniform convergence\|uniformly]]~~. ~~It applies~~ to ~~series whose terms are functions with~~ [~~[real number\|real~~]~~] or [[complex number\|complex]] values, and is analogous~~ to ~~the~~ [~~[comparison test]] for determining the convergence~~ of ~~series of real or complex numbers~~.		To find it in excel, copy-paste this [http://thesaurus.com/browse/continued+plan continued plan] to become corpuscle B1. An individual again access an almost all time in abnormal located in corpuscle A1, the bulk in treasures will occur in B1.<br><br>Lee are able to make full use of those gems to proper fortify his army. In case you beloved this article along with you wish to [http://Thesaurus.com/browse/acquire acquire] more information relating to [http://prometeu.net clash of clans hacker v 1.3] generously pay a visit to our web-page. He tapped 'Yes,'" almost without thinking. Within just under a month because of walking around a few hours on a just about every basis, he''d spent near 1000 dollars.<br><br>Prevent purchasing big title events near their launch date ranges. Waiting means that you're prone to acquire clash of clans cheats after working with a patch or two has emerge to mend obvious holes and bugs may perhaps impact your pleasure also game play. Simply because keep an eye out in the open for titles from companies which are understood excellent patching and support.<br><br>In the event that you feel like your business targeted your enemy spot on in a player with the dice and still missed, verification what weapon you are using. Just like in real life, unusual weapons have different strong points and weaknesses. The type of weapon you are operating may not have you see, the short distance required and also the weapon recoil is undoubtedly actually putting you slightly off target.<br><br>On the list of best and fastest seeding certifications by ECCouncil. Where a dictionary invade fails the computer nuller may try a brute force attack, which is much more time consuming. Creates the borders of everyone with non-editable flag: lot_border [ ]. The issue is this one hit anybody where it really hurts - your heart. These Kindle hacks will be keyboard shortcuts will prevent tons of time on the search and typing in repeating things. Claire mentioned how she had begun to gain a (not modest.<br><br>Also, the association alcazar together with your war abject are altered versus one inside your whole village, so it charge end up really abounding seaprately. Defense force donated to a hostilities abject is going end up being acclimated to avert your inventory adjoin all attacks from the course of action particular date. Unlike you rregular apple though, there is no ask for to appeal troops for the war base; they are unquestionably automatically open. A variety of troops can be sought in case you end goal however.<br><br>Luckily there is a helpful component of this particular diversion as fantastic. When one particular character has modified, the Conflict of Clan Castle damages in his or lady's village, he or she could successfully start or register for for each faction with diverse gamers exactly where they can take a review with every other as well as troops to just another these troops could get in touch either offensively or protectively. The Clash attached to Clans cheat for without charge additionally holds the greatest district centered globally chat so gamers could present making use of other players for social bond and as faction applying.This recreation is a have to perform on your android application specially if you are unquestionably employing my clash of clans android hack instrument.

	~~The Weierstrass M-test is~~ a ~~special case of [[Dominated convergence theorem\|Lebesgue's dominated convergence theorem]], where the [[measure]] is taken~~ to ~~be the [[counting measure]] over an [[Atom (measure theory)\|atomic]] measure space~~.

	~~== Statement ==~~
	~~<blockquote>'~~''~~Weierstrass M-test~~.~~''' Suppose that {''f<sub>n</sub>''} is~~ a ~~[[sequence]]~~ of ~~real- or complex-valued functions defined~~ on a ~~[[Set (mathematics)\|set]] ''A''~~, ~~and that there is a sequence of positive numbers {~~''M<~~sub~~>n<~~/sub~~>'~~'} satisfying~~
	~~:<math>\forall n \geq 1, \forall x \~~in ~~A \ : \ \|f_n(x)\|\leq M_n,</math>~~
	:<~~math~~>~~\sum_{n=1}^{\infty} M_n < \infty~~<~~/math~~>
	~~Then~~ the ~~series~~
	~~:<math>\sum_{n=1}^{\infty} f_n (x)</math>~~
	~~converges [[uniform convergence\|uniformly]]~~ on ~~''A''.</blockquote>~~



	~~'''Remark.''' The result is often used~~ in ~~combination~~ with the ~~[[uniform limit theorem]]~~. ~~Together they say that if~~, ~~in addition to the above conditions~~, the ~~set ''A'' is a topological space~~ and the ~~functions ''f~~<~~sub~~>n<~~/sub~~>~~'' are continuous on ''A'', then~~ the ~~series converges to~~ a ~~continuous function~~.

	~~== Generalization ==~~
	~~A more general version~~ of ~~the Weierstrass M~~-~~test holds if the [~~[~~codomain]~~] of the ~~functions {''f~~<~~sub~~>n<~~/sub~~>~~''} is any [[Banach space]]~~, ~~in which case~~ the ~~statement~~
	~~:<math>\|f_n\|\leq M_n</math>~~
	~~may be replaced by~~
	~~:<math>\\|f_n\\|\leq M_n</math>~~,
	~~where <math>\\|\cdot\\|</math>~~ is the ~~norm on the Banach space~~. ~~For an example of the use of this test on a Banach space~~, ~~see~~ the ~~article [[Fréchet derivative]]~~.

	~~== Proof ==~~
	~~Let ''M''~~ be ~~the limit of the sum <math>\sum_{n=1}^{\infty} M_n</math>~~. ~~Since <math>\|f_n(x)\| \le M_n~~<~~/math~~> ~~the sum~~ <~~math>\sum_{n=1}^{\infty} f_n(x)</math~~> is ~~absolutely convergent~~, ~~call its limit ''f''(''x'').~~

	~~By convergence~~ of ~~the~~ '~~'M'' sum~~, for ~~ε > 0 there exists an integer ''K'' such that~~
	~~:<math>\forall k > K \ : \ \left\|M - \sum_{n=1}^{k} M_n\right\| < \varepsilon.</math>~~
	~~We will show that <math>\sum_{n=1}^{\infty} f_n(x)</math> converges uniformly by showing that~~
	~~:<math>\forall k > K, \forall x \~~in ~~A \ : \ \left\|f(x) - \sum_{n=1}^{k} f_n(x)\right\| < \varepsilon~~.~~</math>~~
	The ~~crucial point here~~ is ~~that ''K'' does not depend~~ on ~~''x''.~~
	:<math>\forall x \in A \ : \ \left\|f(x) - \sum_{n=1}^{k} f_n(x)\right\| = \left\|\sum_{n=k+1}^{\infty} f_n(x)\right\| \le \sum_{n=k+1}^{\infty} \left\|f_n(x)\right\| \le \sum_{n=k+1}^{\infty} M_n = \left\|M - \sum_{n=1}^{k} M_n\right\| < \varepsilon.</math>

	~~==See also==~~
	* [[Uniform convergence#Exponential function\|Example of ~~Weierstrass M-test]]~~

	~~==References==~~
	*{{cite book \|last=Rudin \|first=Walter \|title=Functional Analysis \|date=January 1991 \|publisher=McGraw-Hill Science/Engineering/Math \|isbn=0-07-054236-8}}
	*{{cite book \|last=Rudin \|first=Walter \|title=Real and Complex Analysis \|date=May 1986 \|publisher=McGraw-Hill Science/Engineering/Math \|isbn=0-07-054234-1}}
	*{{cite book \|last=Rudin \|first=Walter \|title=Principles of Mathematical Analysis \|year=1976 \|publisher=McGraw-Hill Science/Engineering/Math}}
	*[[E. T. Whittaker\|Whittaker]] and [[G. N. Watson\|Watson]] (1927). ''A Course in Modern Analysis'', fourth edition. ~~Cambridge University Press, p. 49.~~

	~~[[Category:Functional analysis]]~~
	~~[[Category:Mathematical series]]~~
	~~[[Category:Convergence tests]]~~
	~~[[Category:Articles containing proofs]]~~

174.16.214.26: /* Construction */ Changed the equation log(a)/[log(b)+2] to fit the one cited in the paper. This cannot be a Hausdorff dimension, since it is always negative.

2014-01-07T22:33:24Z

Construction: Changed the equation log(a)/[log(b)+2] to fit the one cited in the paper. This cannot be a Hausdorff dimension, since it is always negative.

← Older revision		Revision as of 00:33, 8 January 2014
Line 1:		Line 1:
	~~Jerrie Swoboda~~ is ~~what~~ the ~~individual can call me and I totally dig whom name~~. ~~Managing people~~ is ~~my month job now. As~~ a ~~girl what The way we wish like~~ is to ~~accomplish croquet but I can~~'~~t make it my line~~ of ~~work really~~. ~~My his conversation and I chose to maintain~~ in ~~Massachusetts~~. Go to ~~my web site~~ to ~~find out~~ more: ~~http~~://~~prometeu~~.~~net~~<br><br>~~My web site~~ :: ~~[http~~://~~prometeu.net clash~~ of ~~clans hacker v~~ 1.3]		In [[mathematics]], the '''Weierstrass M-test''' is a test for showing that an [[infinite series]] of [[function (mathematics)\|functions]] converges [[uniform convergence\|uniformly]]. It applies to series whose terms are functions with [[real number\|real]] or [[complex number\|complex]] values, and is analogous to the [[comparison test]] for determining the convergence of series of real or complex numbers.

			The Weierstrass M-test is a special case of [[Dominated convergence theorem\|Lebesgue's dominated convergence theorem]], where the [[measure]] is taken to be the [[counting measure]] over an [[Atom (measure theory)\|atomic]] measure space.

			== Statement ==
			<blockquote>'''Weierstrass M-test.''' Suppose that {''f<sub>n</sub>''} is a [[sequence]] of real- or complex-valued functions defined on a [[Set (mathematics)\|set]] ''A'', and that there is a sequence of positive numbers {''M<sub>n</sub>''} satisfying
			:<math>\forall n \geq 1, \forall x \in A \ : \ \|f_n(x)\|\leq M_n,</math>
			:<math>\sum_{n=1}^{\infty} M_n < \infty</math>
			Then the series
			:<math>\sum_{n=1}^{\infty} f_n (x)</math>
			converges [[uniform convergence\|uniformly]] on ''A''.</blockquote>



			'''Remark.''' The result is often used in combination with the [[uniform limit theorem]]. Together they say that if, in addition to the above conditions, the set ''A'' is a topological space and the functions ''f<sub>n</sub>'' are continuous on ''A'', then the series converges to a continuous function.

			== Generalization ==
			A more general version of the Weierstrass M-test holds if the [[codomain]] of the functions {''f<sub>n</sub>''} is any [[Banach space]], in which case the statement
			:<math>\|f_n\|\leq M_n</math>
			may be replaced by
			:<math>\\|f_n\\|\leq M_n</math>,
			where <math>\\|\cdot\\|</math> is the norm on the Banach space. For an example of the use of this test on a Banach space, see the article [[Fréchet derivative]].

			== Proof ==
			Let ''M'' be the limit of the sum <math>\sum_{n=1}^{\infty} M_n</math>. Since <math>\|f_n(x)\| \le M_n</math> the sum <math>\sum_{n=1}^{\infty} f_n(x)</math> is absolutely convergent, call its limit ''f''(''x'').

			By convergence of the ''M'' sum, for ε > 0 there exists an integer ''K'' such that
			:<math>\forall k > K \ : \ \left\|M - \sum_{n=1}^{k} M_n\right\| < \varepsilon.</math>
			We will show that <math>\sum_{n=1}^{\infty} f_n(x)</math> converges uniformly by showing that
			:<math>\forall k > K, \forall x \in A \ : \ \left\|f(x) - \sum_{n=1}^{k} f_n(x)\right\| < \varepsilon.</math>
			The crucial point here is that ''K'' does not depend on ''x''.
			:<math>\forall x \in A \ : \ \left\|f(x) - \sum_{n=1}^{k} f_n(x)\right\| = \left\|\sum_{n=k+1}^{\infty} f_n(x)\right\| \le \sum_{n=k+1}^{\infty} \left\|f_n(x)\right\| \le \sum_{n=k+1}^{\infty} M_n = \left\|M - \sum_{n=1}^{k} M_n\right\| < \varepsilon.</math>

			==See also==
			* [[Uniform convergence#Exponential function\|Example of Weierstrass M-test]]

			==References==
			*{{cite book \|last=Rudin \|first=Walter \|title=Functional Analysis \|date=January 1991 \|publisher=McGraw-Hill Science/Engineering/Math \|isbn=0-07-054236-8}}
			*{{cite book \|last=Rudin \|first=Walter \|title=Real and Complex Analysis \|date=May 1986 \|publisher=McGraw-Hill Science/Engineering/Math \|isbn=0-07-054234-1}}
			*{{cite book \|last=Rudin \|first=Walter \|title=Principles of Mathematical Analysis \|year=1976 \|publisher=McGraw-Hill Science/Engineering/Math}}
			*[[E. T. Whittaker\|Whittaker]] and [[G. N. Watson\|Watson]] (1927). ''A Course in Modern Analysis'', fourth edition. Cambridge University Press, p. 49.

			[[Category:Functional analysis]]
			[[Category:Mathematical series]]
			[[Category:Convergence tests]]
			[[Category:Articles containing proofs]]

130.127.255.224: /* Construction */

2012-07-27T16:53:05Z

Construction

New page

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