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Topic: Divergent series

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	Divergent series - Wikipedia, the free encyclopedia
		That is, divergent series and convergent series are antonyms.
		The divergence of the harmonic series was proved by the medieval mathematician Nicole Oresme.
		The subject of divergent series, as a domain of mathematical analysis, is primarily concerned with explicit and natural techniques such as Abel summation, Cesàro summation and Borel summation, and their relationships.
	en.wikipedia.org /wiki/Divergent_series (371 words)

	ipedia.com: Divergence Article (Site not responding. Last check: 2007-10-21)
		In vector calculus, the divergence is an operator that measures a vector field's tendency to originate from or converge upon a given point.
		In physical terms, the divergence of a vector field is the extent to which the vector field flow behaves like a source or a sink at a given point.
		The Laplacian of a scalar field is the divergence of the field's gradient.
	www.ipedia.com /divergence.html (605 words)

	SERIES (a Latin word f... - Online Information article about SERIES (a Latin word f... (Site not responding. Last check: 2007-10-21)
		CH2, and in the form isologous series, applied to hydrocarbons and their derivatives which differ in empirical composition by a multiple of H2; it is also used in the form isomorphous series to denote elements related isomorphously.
		The series whose general term is of the form Kan+..(n), where 0(n) is a rational integral algebraic function of degree r, is a recurring series whose scale of relation is (I—ax) (i —x)1, but the general term of this series** may be obtained by another method.
		Such a series may be absolutely convergent and the sum is then independent of the order of the terms and is equal to the sums of the two series us-dui+u2+...
	encyclopedia.jrank.org /SCY_SHA/SERIES_a_Latin_word_from_serere.html (5488 words)

	SERIES - LoveToKnow Article on SERIES (Site not responding. Last check: 2007-10-21)
		It follows that the general term of a recurring series is of the form ~(n)a~, where ~(n) is a rational integral algebraic function of is, and a is independent of n.
		The series whose general term is of the form Ka+.~.(n), where ~(n) is a rational integral algebraic function of degree r, is a recurring series whose scale of relation is (Iax) (I _x)~~i, but the general term of this series may be obtained by another method.
		Such a series is said to be absolutely convergent when the series of moduli of is,,, i.e., ~(p~i+q,,2)l, is convergent; this is sufficient but not necessary for the separate convergence of the p and q series.
	www.1911ency.org /S/SE/SERIES.htm (5830 words)

	In the folding of divergent series... (Site not responding. Last check: 2007-10-21)
		Deleuze, however, notes that the question is not so much one of seriality as of the cognition of seriality: to be in a series is not equivalent to perceiving the seriality of the series, let alone its relations to other series.
		The series of the world exists within the monad, but it is chosen by God, and the harmony that exists among monads is preestablished by God in reference to a limit of seriality that is wholly extrinsic to the series.
		Divergences create branchings, as when, in a hypertext fiction, one can choose from a number of plot developments of varying degrees of compatibility, each path containing yet more divergences within itself, the entire text branching endlessly into multiple narratives, some of which will be incompossible with each other.
	home.earthlink.net /~dysthymic/brock99.html (3177 words)

	Talk:Divergent series - Wikipedia, the free encyclopedia
		There is a quick test to prove an infinite series is convergent or divergent.
		Reason (2) is that this really belongs on the basic infinite series page.
		This is the first of the convergence criteria given at infinite series, in fact.
	en.wikipedia.org /wiki/Talk:Divergent_series (159 words)

	[No title]
		A geometric series is a series which follows the pattern, EMBED "Equation" \* mergeformat where EMBED "Equation" \* mergeformat is the initial term and EMBED "Equation" \* mergeformat is a ratio (i.e., EMBED Equation.DSMT4 , EMBED Equation.DSMT4 , EMBED Equation.DSMT4 , etc.).
		SERIES WITH BOTH POSITIVE AND NEGATIVE TERMS The next sets of series tests are those that apply to series with both negative and positive terms (known as alternating series).
		An alternating series is a series of the form EMBED "Equation" "Word Object6" \* mergeformat , in other words, a series whose terms alternate between positive and negative.
	www.utexas.edu /student/utlc/lrnres/mstc/a315/calchandouts/infiniteseries.doc (2062 words)

	Formal-Numeric Series (Site not responding. Last check: 2007-10-21)
		The importance of this for conditionally convergent numerical series (such as 1 - 1/2 + 1/3 - 1/4 +...) is obvious, but this condition is also significant for associations between formal-numeric divergent series.
		On this basis it's clear that the natural series is "equivalent" to any other divergent series, such as the geometric series of powers of 2, so this kind of "formal equivalence" is not particularly interesting.
		In general, since the nth partial sum of the natural sequence is n(n+1)/2, and the mth partial sum of the geometric sequence 1 + b + b^2 +...
	www.mathpages.com /home/kmath500.htm (448 words)

	PlanetMath: Abel summability
		It extends the usual definition of the sum of a series, and gives a way of summing up certain divergent series.
		The standard example of a divergent series that is nonetheless Abel summable is the alternating series
		In all cases, the conclusion is that a certain series or a certain integral is convergent.
	planetmath.org /encyclopedia/AbelianTheorem.html (387 words)

	PlanetMath: slower divergent series
		is a diverging series with positive terms, then one can always form another diverging series
		This series is however divergent, because the sum of its
		This is version 9 of slower divergent series, born on 2005-03-19, modified 2005-03-21.
	planetmath.org /encyclopedia/SlowerDivergentSeries.html (104 words)

	Convergent Series
		Thus a series is convergent if and only if it's sequence of partial sums is convergent.
		The limit of the sequence of partial sums is the sum of the series.
		A series which is not convergent, is a divergent series.
	www.maths.abdn.ac.uk /~igc/tch/ma2001/notes/node49.html (262 words)

	Search Results for Series (Site not responding. Last check: 2007-10-21)
		In a remarkable series of papers which appeared in the Annals of Mathematics he investigated a differential group of order n was he defined as a power series, with certain properties, in two sets of n indeterminates and their derivatives of various orders.
		He also contributed to the study of divergent series, a topic which interested him early in his career, and we should note that in his work on mathematical physics he was a staunch follower of Maxwell.
		Bessel functions appear as coefficients in the series expansion of the indirect perturbation of a planet, that is the motion caused by the motion of the Sun caused by the perturbing body.
	www-gap.dcs.st-and.ac.uk /~history/Search/historysearch.cgi?SUGGESTION=Series&CONTEXT=1 (16418 words)

	Explicit Remainder Estimates
		In the context of the summation of divergent series, the books by Borel [52] and Doetsch [53] and the review article by Thomann [54] are of interest, since the connection between factorial series and summability is discussed there.
		Summation methods, which are closely related to factorial series, are discussed by Gunson and Ng [55,56], and the analytical extension of functions defined by factorial series is discussed by Hughes [57].
		Power series converge in circles, which may shrink to a single point or extend to contain the whole complex plane, whereas factorial series converge according to Landau [62] in half-planes.
	www.apmaths.uwo.ca /~rcorless/AM563/NOTES/report/node4.html (1073 words)

	taylor2sol
		Decide where the new series is centered at and then use the ratio test to determine the radius of convergence of the new series.
		Then determine the convergence or divergence of this series at the endpoints of the interval of convergence.
		Since the alternating series is also a geometric series we can determine its actual sum without the estimation theorem for alternating series.
	www-math.cudenver.edu /~rrosterm/taylor2sol/taylor2sol.html (657 words)

	Convergence Criteria
		But, on the other hand, the harmonic series is divergent while the hyper harmonic series is convergent.
		Note that this is essentially the hyperharmonic series version of an improper integrals result, namely (5.3.7).
		This series is convergent (6.2.7) but taking the absolute value of each term would make it the harmonic series, which is divergent (6.2.2).
	www.math.uconn.edu /~corluy/calculus/lecturenotes/node22.html (477 words)

	Divergent series of the Fibonacci numbers
		A series is a sum of terms specified by some rule.
		Fibonacci (1170 - 1240) discovered a sequence of integers in which each number is equal to the sum of the preceding two introducing it in terms of modeling a breeding population of rabbits.
		We assume that all divergent series are convergent.
	milan.milanovic.org /math/english/divergent/divergent.html (377 words)

	A Radical Approach to Real Analysis
		Cauchy tried to banish the practice of assigning values to series that do not converge.As Daniel Bernoulli showed (see section 2.6), there are traps that are easy to fall into when we attempt to assign values to divergent series.
		A divergent series cannot give us an arbitarily close approximation to a given value, but it might be able to give us an approximation that is good enough for our purposes.
		H. Hardy published his classic book, Divergent Series, on methods of handling and assigning values to such series as well as on demonstrations of their usefulness.
	www.macalester.edu /aratra/chapt2/chapt2_2_1.html (248 words)

	15.4 Divergent Series (Site not responding. Last check: 2007-10-21)
		For some series, we are interested in the terms, not their convergence.
		The individual terms may count something we want to count, and fact that the function defined by the series diverges is of little interest to us.
		There are some power series that we can define even where they diverge, especially if we allow our variable to be complex.
	www-math.mit.edu /~djk/calculus_beginners/chapter15/section04.html (272 words)

	Stieltjes Series
		These two transformations are based upon the remainder estimate (5.3), which corresponds to the first term of the series not contained in the partial sum.
		The classic example of a Stieltjes function with a strongly divergent Stieltjes series is the Euler integral (2.6) and its associated asymptotic series, the Euler series (2.7).
		Stieltjes functions and their associated Stieltjes series are very important in the theory of divergent series, since they possess a highly developed representation and convergence theory [25,71,26,72,73,74].
	www.apmaths.uwo.ca /~rcorless/AM563/NOTES/report/node6.html (644 words)

	ReuvenLax (Site not responding. Last check: 2007-10-21)
		A convergence definition was probably given as the convergence of the sequence of partial sums of the series.
		And, oh what a lie that is. Divergent series have held mathematicians interest for quite some while.
		Any series that is convergent using the calculus-course definition of convergence must also converge using our new definition, and furthermore must converge to the same value.
	blogs.msdn.com /reuvenlax/default.aspx (8488 words)

	Practice Quiz 2 (Site not responding. Last check: 2007-10-21)
		Ignoring the 2, you are left with the series (2/3)^n which is a geometric one with r=2/3 and thus converges.
		In an exam, compare this series with the geometric series (2/3)^n and show that it is smaller.
		Our series is an alternating one, so we have to see if it converges.
	www.math.temple.edu /~visnjic/Math86OL/Quiz5Prac.html (398 words)

	Math 222 (Site not responding. Last check: 2007-10-21)
		This is an exploration of one aspect of infinite series.
		Briefly explain what it means for one divergent series to be “smaller” than another divergent series (after all, both series sum to infinity).
		Without showing any work, write down a divergent series that would be smaller than any of the series of the form
	www.roanoke.edu /staff/minton/221h3.htm (243 words)

	Calculus II (Math 2414) - Series & Sequences - Integral Test (Site not responding. Last check: 2007-10-21)
		With the harmonic series this was all that we needed to say that the series was divergent. With this series however, this isn’t quite enough. For instance
		So, once again we were able to relate a series to an improper integral (that we could compute) and the series and the integral had the same convergence.
		It is important to note before leaving this section that in order to use the Integral Test the series terms MUST be positive. If they are negative then the test doesn’t work. Also remember that the test only determines the convergence of a series and does NOT give the value of the series.
	tutorial.math.lamar.edu /AllBrowsers/2414/IntegralTest.asp (975 words)

	[No title] (Site not responding. Last check: 2007-10-21)
		Slowly convergent series and sequences as well as divergent series occur quite frequently in the mathematical treatment of scientific problems.
		In this report, a large number of mainly nonlinear sequence transformations for the acceleration of convergence and the summation of divergent series are discussed.
		Finally, the performance of the sequence transformations of this report are tested by applying them to certain slowly convergent and divergent series, which are hopefully realistic models for a large part of the slowly convergent or divergent series that can occur in scientific problems and in applied mathematics.
	celestial.eprints.org /cgi-bin/oaia2/arXiv.org?verb=GetRecord&identifier=oai:arXiv.org:math/0306302&metadataPrefix=oai_dc (173 words)

	A Radical Approach to Real Analysis
		Euler worked with divergent series and, as we saw in section 2.2, determined the value from the genesis of the series.
		There is a difficulty with this point of view that was exposed by Jean Bernoulli's son Daniel (1700–1782) in 1772: different machinery can give rise to the same series with different values.
		On the other hand, to simply discard all divergent series is to lose those, like the error term in Stirling's formula, that are truly useful.
	www.macalester.edu /aratra/chapt2/chapt2_6a.html (170 words)

	The Particular Case of Positive Series
		The last result on positive series may be the most useful of all.
		Hence, by the Limit-test, we deduce the convergence of the series
		is divergent, the limit-test implies that the series
	www.sosmath.com /calculus/series/poseries/poseries.html (421 words)

	Calculus II (Math 2414) - Series & Sequences - Ratio Test
		Before proceeding with the test let’s do a quick reminder of factorials. This test will be particularly useful for series that contain factorials (and we will see some in the applications) so let’s make sure we can deal with them before we run into them in an example.
		the series may be divergent, convergent, or absolutely convergent.
		which means the ratio test is no good for determining the convergence of this series. We will need to resort to another test for this series. This series is an alternating series and so let’s check the two conditions from that test.
	tutorial.math.lamar.edu /AllBrowsers/2414/RatioTest.asp (897 words)

	Abel's theorem - meaning of word
		In real analysis, '''Abel's theorem''' for power series with non-negative coefficients relates a limit of a power series to the sum of its coefficients.
		''z'') approaches 1 from below, even in cases where the radius of convergence, ''R'', of the power series is equal to 1 and we cannot be sure whether the limit should be finite or not.
		The field of divergent series, and their summation methods, contains many theorems ''of abelian type'' and ''of tauberian type''.
	wordsonline.org /Abel%27s_theorem (239 words)

	quizzf018sol
		Determine the convergence or divergence of the following series.
		Solution:Rewrite the series as a convergent geometric series with
		Solution:Rewrite the series as the difference between two series and evaluate each series individually.
	www-math.cudenver.edu /~rrosterm/quizzf018sol/quizzf018sol.html (140 words)

	[No title]
		This is one of the standard examples of divergent series, treated in detail by Laguerre in 1879, and discussed in just about every survey of divergent series.
		Of course, you are taking liberties by assigning the label "f(z)" both to this function and to the divergent series.
		We should really put the word "sum" in quotation marks when referring to a valuation of a divergent series.
	www.math.niu.edu /~rusin/known-math/00_incoming/divseq (882 words)

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