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Topic: Alternating series

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	Rearranging The Alternating Harmonic Series (Intro)
		Conditionally convergent series are those series that converge as written, but do not converge when each of their terms is replaced by the corresponding absolute value.
		Students see the usefulness of studying absolutely convergent series since most convergence tests are for positive series, but to them conditionally convergent series seem to exist simply to provide good test questions for the instructor.
		It is clear, however, that even with such a simple example as the alternating harmonic series one cannot hope for a closed form solution to the problem of rearranging it to sum to an arbitrary real number.
	ecademy.agnesscott.edu /~lriddle/series/rear.htm (552 words)

	Series (mathematics) - Wikipedia, the free encyclopedia
		Series may be finite, or infinite; in the first case they may be handled with elementary algebra, but infinite series require tools from mathematical analysis if they are to be applied in anything more than a tentative way.
		Series for the expansion of sines and cosines, of multiple arcs in powers of the sine and cosine of the arc had been treated by Jakob Bernoulli (1702) and his brother Johann Bernoulli (1701) and still earlier by Viète.
		Formal power series are also used in combinatorics to describe and study sequences that are otherwise difficult to handle; this is the method of generating functions.
	en.wikipedia.org /wiki/Series_(mathematics) (1781 words)

	Alternating series - Wikipedia, the free encyclopedia
		In mathematics, an alternating series is an infinite series of the form
		A sufficient condition for the series to converge is that it converges absolutely.
		A conditionally convergent series is an infinite series that converges, but does not converge absolutely.
	en.wikipedia.org /wiki/Alternating_series (232 words)

	PlanetMath: alternating series test
		This test provides a sufficient (but not necessary) condition for the convergence of an alternating series, and is therefore often used as a simple first test for convergence of such series.
		is necessary for convergence of an alternating series.
		This is version 12 of alternating series test, born on 2002-02-24, modified 2004-11-24.
	planetmath.org /encyclopedia/AlternatingSeriesTest.html (135 words)

	Alternating series -- Facts, Info, and Encyclopedia article (Site not responding. Last check: 2007-10-24)
		In (A science (or group of related sciences) dealing with the logic of quantity and shape and arrangement) mathematics, an alternating series is an (additional info and facts about infinite series) infinite series of the form
		A sufficient condition for the (Similar things placed in order or happening one after another) series to converge is that it (additional info and facts about converges absolutely) converges absolutely.
		A broader test for convergence of an alternating series is the Cauchy criterion: if the sequence is monotone decreasing and tends to zero, then the series
	www.absoluteastronomy.com /encyclopedia/a/al/alternating_series.htm (247 words)

	PlanetMath: alternating series
		An alternating series is a series of the form
		Loosely, this is just a series where the terms ``alternate'' between positive and negative.
		This is version 3 of alternating series, born on 2002-02-24, modified 2005-03-25.
	planetmath.org /encyclopedia/AlternatingSeries.html (65 words)

	Math Forum - Ask Dr. Math
		That is, twice the harmonic series, minus the harmonic series.
		For the alternating series test, you need that for some term n, from that term on it's ALWAYS decreasing.
		Alternating series that aren't absolutely convergent, but do converge, are very tricky to work with!
	mathforum.org /library/drmath/view/56966.html (679 words)

	S.O.S. Mathematics CyberBoard :: View topic - Alternating Series Test
		I hope it is possible because I would like to estimate the sum by using the Alternating Series Estimation Theorem, which I can't use until I know that the given series is convergent by the A.S.T. I m having trouble showing the limit as n approaches infinity of Bn (b sub-n) = 0.
		Do you know what the alternating series test (Leibniz' test) is? It's pretty straightforward to verify the conditions that has to be met...
		Don't forget that the alternating series test has a monotonicity condition as well.
	www.sosmath.com /CBB/viewtopic.php?t=15223&highlight= (561 words)

	Series -- Types and Tests (PRIME)
		This important series should be thought of as a function in x for all x in the radius of convergence.
		A series converges when its sequence of partial sums converges, that is, if the sequence of values given by the first term, then the sum of the first two terms, then the sum of the first three terms, etc., converges as a sequence.
		A series is said to converge absolutely if the series still converges when all of the terms of the series are made non-negative (by taking their absolute value).
	www.mathacademy.com /pr/prime/articles/serie (681 words)

	8 (Site not responding. Last check: 2007-10-24)
		series is a series with on of the two following forms:
		is not satisfied, the series may or may not converge.
		(The alternating harmonic series is conditionally convergent.)
	www.gpc.edu /~jcraig/calc2_ch8/8s4_other_tests.htm (169 words)

	[No title]
		Now we are going to investigate the alternating series, and determine the convergence and divergence of these series.
		A series in which the terms are alternating between positive and negative is called an alternating series.
		The alternating series test will be used in the next topic, which is absolute and conditional convergence of series.
	faculty.eicc.edu /bwood/ma155supplemental/supplemental24.htm (467 words)

	Calculus II (Math 2414) - Series & Sequences - Alternating Series Test (Site not responding. Last check: 2007-10-24)
		The last two tests that we looked at for series convergence have required that all the terms in the series be positive. Of course there are many series out there that have negative terms in them and so we now need to start looking at tests for these kinds of series.
		The series from the previous example is sometimes called the Alternating Harmonic Series.
		The two conditions of the test are met and so by the Alternating Series Test the series is convergent.
	tutorial.math.lamar.edu /AllBrowsers/2414/AlternatingSeries.asp (805 words)

	The commutative law doesn't hold for certain series (Site not responding. Last check: 2007-10-24)
		It converges by the alternating series test, but the sum of its absolute values is the harmonic series, which diverges.
		Therefore, the alternating harmonic series is conditionally convergent.
		The error estimate for the alternating series test tells you that S is between 1/2 and 1 (in fact, it equals
	www.math.tamu.edu /~tom.vogel/gallery/node10.html (231 words)

	4.1. Series and Convergence
		Note that while a series is the result of an infinite addition - which we do not yet know how to handle - each partial sum is the sum of finitely many terms only.
		Hence, there are different modes of convergence: one mode that applies to series with positive terms, and another mode that applies to series whose terms may be negative and positive.
		Then any rearrangement of terms in that series results in a new series that is also absolutely convergent to the same limit.
	web01.shu.edu /projects/reals/numser/series.html (823 words)

	[No title]
		This means that if the positive term series converges, then both the positive term series and the alternating series will converge.
		This means that the positive term series diverges, but the alternating series converges.
		The third condition holds, therefore the alternating series converges, and the given series converges conditionally.
	faculty.eicc.edu /bwood/ma155supplemental/supplemental25.htm (870 words)

	altseries.html
		Approximating the sum of a series that converges by the Alternating Series Test.
		The error made when using the nth partial sum of a series that converges by the Alternating series test as the sum of the entire series is less than the absolute value of the (n+1)st term.
		Note that each term of the series is defined as a function of n.
	www.wfu.edu /users/ekh/maple/altseries/altseries1.html (271 words)

	Calculus II (Math 2414) - Series & Sequences - Absolute Convergence (Site not responding. Last check: 2007-10-24)
		It is this fact that makes absolute convergence a “stronger” type of convergence. Series that are absolutely convergent are guaranteed to be convergent. However, series that are convergent may or may not be absolutely convergent.
		Therefore, this series is not absolutely convergent. It is however conditionally convergent since the series itself does converge.
		This series is convergent by the p-series test and so the series is absolute convergent. Note that this does say as well that it’s a convergent series.
	tutorial.math.lamar.edu /AllBrowsers/2414/AbsoluteConvergence.asp (369 words)

	Definition: Alternating Harmonic Series (Site not responding. Last check: 2007-10-24)
		the series of absolute values is a p-series with p = 1, and diverges by the p-series test.
		The original series converges, because it is an alternating series, and the alternating series test applies easily.
		However, here is a more elementary proof of the convergence of the alternating harmonic series.
	web01.shu.edu /projects/reals/numser/proofs/altharm.html (162 words)

	Conditional Convergence and its pathologies (Site not responding. Last check: 2007-10-24)
		Conditionally convergent series will have a a subsequence of positive terms and a subsequence of negative terms which both give series which diverge.
		These can be used to produce a rearrangement of the series which converges to any value we wish.
		Since the rearrangement of series is an infinite analog of the commutative law, we must consider this behavior pathological.
	www.iwu.edu /~lstout/series/node14.html (172 words)

	alternating (Site not responding. Last check: 2007-10-24)
		Here is a new criterion for testing when a series is alternating.
		as the sum of a series and then carefully show that the resulting series is alternating (or at least that it's alternating after the first few terms).
		The terms of the series certainly look to be alternating, at least after the first term.
	math.columbia.edu /~loftin/calc2aspring03/alternating/alternating.html (273 words)

	Solutions to Homework #4
		So the series is absolutely convergent (and therefore convergent.
		Consider the series whose terms are the absolute values o the terms of the given series.
		So the series is absolutely convergent by the Ratio Test.
	www.rit.edu /~dansma/253HMK6/Homework6.htm (74 words)

	Math Forum - Ask Dr. Math (Site not responding. Last check: 2007-10-24)
		Date: 02/23/98 at 21:57:19 From: Harris Welt Subject: convergence of an alternating series Dear Dr. Math, I'm trying to figure out whether or not this series is convergent.
		Date: 02/24/98 at 09:34:25 From: Doctor Anthony Subject: Re: convergence of an alternating series For an alternating series, you will ALWAYS have convergence if the limit of u(r) as r -> infinity is zero.
		n 1 You can see why the alternating series converges if you plot S1, S2, S3, etc on a number line.
	mathforum.org /library/drmath/view/56921.html (247 words)

	Programmed tutorial: Alternating Series: Convergence or Divergence: Example 1
		Programmed tutorial: Alternating Series: Convergence or Divergence: Example 1
		Since it is an alternating series, we can try to show that
		Both conditions are satisfied for this alternating series to converge.
	www.jtaylor1142001.net /calcjat/Solutions/Series/Alternating-Series/Alternating-Series-1/Alternating-Series-1-Layers.htm (286 words)

	Alternating Series Worksheet (Site not responding. Last check: 2007-10-24)
		(e) Use partial sums to find the sum of the series accurately to 6 significant digits.
		(d) Determine whether this series converge absolutely or conditionally.
		Based on these examples, which type of series would you say converges faster, an absolutely or conditionally convergent series?
	www.cems.uvm.edu /~read/math22/hw_alternating-series (471 words)

	Alternating series and good approximations
		The alternating series test is particularly nice because it does give us an error estimate:
		diverges, even though it is an alternating series because n does not go to 0.
		First find the radius of convergence using the ratio test, then use limit comparison with a p-series or the alternating series test to see what happens at the endpoints.
	www.iwu.edu /~lstout/series/node13.html (241 words)

	Series Convergence Tests
		If the alternating series converges, then the remainder R
		of the Taylor series (where S is the exact sum of the infinite series and S
		is the sum of the first N terms of the series) is equal to (1/(n+1)!) f
	www.math2.org /math/expansion/tests.htm (320 words)

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