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Topic: Alternating series test

	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 (136 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/AlternatingSeriesTest2.html (65 words)

	[No title]
		A series in which the terms are alternating between positive and negative is called an alternating series.
		Therefore this series converges by the alternating series test.
		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)

	[No title] (Site not responding. Last check: 2007-10-16)
		To use this test, the series being evaluated must have an absolute value less than or equal to the absolute value of some other series, either for any n or for any n beginning at some finite point and going to infinity.
		Geometric Series Test — If a sequence is in the form of some constant times another constant that is raised to the power of some linear function of n, then the series is geometric.
		This test states that an alternating series will converge if the absolute value of the terms is decreasing, and if the limit as the sequence goes to an arbitrarily large number is zero.
	www.stolaf.edu /people/mceacher/work/math128b/definitions.doc (668 words)

	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)

	Alternating Series Test (Site not responding. Last check: 2007-10-16)
		In fact, when checking for absolute convergence the term 'alternating series' is meaningless.
		It is important that the series truly alternates, that is each positive term is followed by a negative one, and visa versa.
		If that is not the case, the alternating series test does not apply (while Abel's Test may still work).
	pirate.shu.edu /~wachsmut/ira/numser/t_alter.html (85 words)

	SparkNotes: Series: Terms
		Comparison Test - A series with positive terms converges if there is another series with all terms greater or equal which is known to converge.
		Similarly, a series with positive terms diverges if there is another series with all terms lesser or equal which diverges.
		Series - A sum of the elements in a sequence.
	www.sparknotes.com /math/calcbc2/series/terms.html (357 words)

	Series
		To show convergence, you must find a series known to converge that is greater than the given series.
		Series diverges if any one of the conditions is not met.
		For example, the alternating harmonic series converges by the alternating series test but the nonnegative harmonic series diverges.
	www.bucks.edu /~farberb/series.htm (305 words)

	The commutative law doesn't hold for certain series (Site not responding. Last check: 2007-10-16)
		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)

	SparkNotes: Series: Introduction and Summary
		The primary method for determining whether or not a series converges is called the comparison test.
		Alternating series come with their own test for convergence.
		Power series contain a variable; whether or not they converge depends upon what value is substituted for the variable.
	www.sparknotes.com /math/calcbc2/series/summary.html (237 words)

	Calculus II (Math 2414) - Series & Sequences - Alternating Series Test (Site not responding. Last check: 2007-10-16)
		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.
		There are a couple of things to note about this test. First, unlike the Integral Test and the Comparison/Limit Comparison Test, this test will only tell us when a series converges and not if a series will diverge.
		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)

	Question 1 (Site not responding. Last check: 2007-10-16)
		This series converges by the alternating series test, but to get its sum exactly, we have to notice that it is geometric, with first term
		This series converges by the alternating series test:
		So both series either converge of diverge, by the limit comparison test.
	www.math.pitt.edu /~sparling/024/23024/23024q5s/node2.html (151 words)

	Alternating series test - Wikipedia, the free encyclopedia
		The alternating series test is a method used to prove that infinite series of terms converge.
		It was discovered by Gottfried Leibniz and is sometimes known as Leibniz's test or Leibniz criterion.
		are positive or 0, is called an alternating series.
	en.wikipedia.org /wiki/Alternating_series_test (153 words)

	lab10.htm (Site not responding. Last check: 2007-10-16)
		In section 10.5 you were introduced to the alternating series test.
		In the cases where the original series is a convergent alternating series, estimate the error that the sum of the first 100 terms will give.
		Again, if the series is convergent, use Maple to approximate the sum by finding the sum of the first 100 terms, and where possible estimate the error that this sum provides.
	www.uwec.edu /math/Calculus/215-Spring2001/labs/lab10/lab10.htm (284 words)

	Absolute Convergence and Conditional Convergence
		Alternating Series Test may tell you that the series converges.
		One approach you might take to series with negative terms is to force all the negative terms to be positive by taking absolute values.
		Forcing all the terms to be positive should make it more difficult for a series to converge, since you lose the benefit of having negative terms cancelling with positive terms (which might keep the partial sums from blowing up).
	marauder.millersville.edu /~bikenaga/calculus/absconv/absconv.html (563 words)

	Alternating Series Test (Site not responding. Last check: 2007-10-16)
		In fact, when checking for absolute convergence the term 'alternating series' is meaningless.
		It is important that the series truly alternates, that is each positive term is followed by a negative one, and visa versa.
		If that is not the case, the alternating series test does not apply (while Abel's Test may still work).
	web01.shu.edu /projects/reals/numser/t_alter.html (85 words)

	Convergent series - Wikipedia, the free encyclopedia
		In mathematics, a series is the sum of the terms of a sequence of numbers.
		The Riemann series theorem states that if a series converges conditionally, it is possible to rearrange the terms of the series in such a way that the series converges to any value, or even diverges.
		There is an analogue of the comparison test for infinite series of functions called the Weierstrass M-test.
	en.wikipedia.org /wiki/Convergent_series (448 words)

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	www.50cycles.com (940 words)

	Conditional Convergence
		What matters is that the sign alternates; once it is positive, the next one is negative and so on....
		Using the Bertrand Series Test, we conclude that it is divergent.
		Therefore, all the Alternating Series Test assumptions are satisfied.
	www.sosmath.com /calculus/series/conditionnal/conditionnal.html (409 words)

	Solutions to Homework #4
		Þ The series converges by the Alternating Series
		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)

	Question 11 (Site not responding. Last check: 2007-10-16)
		, so the terms of the series are alternately positive and negative.
		So the conditions for the alternating series test apply and the series converges.
		This series alternates in sign, is decreasing and has limit zero, so the series converges by the Alternating Series Test.
	www.math.pitt.edu /~sparling/23023/23023finps/node12.html (83 words)

	Mathwords: Alternating Series Remainder
		A quantity that measures how accurately the nth partial sum of an alternating series estimates the sum of the series.
		If the series converges to S by the alternating series test, then the remainder
		Remainder of a series, convergence tests, divergent series
	www.mathwords.com /a/alternating_series_remainder.htm (80 words)

	Series Convergence Tests (Site not responding. Last check: 2007-10-16)
		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.math.com /tables/expansion/tests.htm (322 words)

	[No title]
		Use the integral test to determine convergence or divergence.
		Use the ratio test to determine convergence or divergence.
		Use the alternating series test to determine convergence.
	math.arizona.edu /~winkel/129test3review.doc (1085 words)

	11.3 Alternating Series (Site not responding. Last check: 2007-10-16)
		The alternating series test has obvious generalizations for series such as
		We will now explicitly calculate the limit of this series using a few ideas that are not justified by results proved in this course.
		Determine whether or not the following series converge.
	www.reed.edu /~mayer/html2/node30.html (78 words)

	256st1
		Know the geometric series, when they converge, and the sum.
		Practice the problems in this section, and be able to apply the correct test to determine if a series converges or diverges.
		Be able to apply a series test (likely the ratio test) to determine where a power series converges
	www.math.cmu.edu /~tim/256s04/256st1/256st1.html (213 words)

	Alternating series and good approximations
		While the several tests we have given so far for convergence give us useful information, they do not, in general, tell us what our error is if we truncate the series.
		The alternating series test is particularly nice because it does give us an error estimate:
		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)

	Calculus II Assignments
		Read section 12.2 on series and series convergence, geometric series, Divergence test, and Series properties or rules.
		Read ahead in section 12.5 about the Alternating Series Test, and estimating the sum of an alternating series.
		Read section 12.4 on the alternating series test, and how to estimate the sum of a series converging by the alternating series test.
	www.wfu.edu /~ekh/mth112/Fall01/assignments/assign.html (1757 words)

	Calculus II (Summer '04) (Site not responding. Last check: 2007-10-16)
		Series (11.1-11.5): divergence test, integral test, comparison test, alternating series test.
		Power Series (11.8-11.10): power series, interval and radius of convergence, Taylor series.
		For power series use the last homework HW5.
	www.math.columbia.edu /~laza/teaching/calc2s.htm (533 words)

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