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Topic: Convergent series

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	Series - LoveToKnow 1911
		The series i - i + I - I+..., where S n is unity or zero, according as n is odd or even, is an example of an oscillating series.
		In the case of a power series there is a quantity R such that the series converges if I z < R, and diverges if I z I > R. A circle described with the origin as centre and radius R is called the circle of convergence.
		The circle of convergence may be of infinite radius as in the case of the series for sin z, viz.
	www.1911encyclopedia.org /Series (4894 words)

	Springer Online Reference Works
		A sequence of elements (called the terms of the given series) of some linear topological space and a certain infinite set of their partial sums (called the partial sums of the series) for which the notion of a limit is defined.
		A condition for the convergence of a series which does not use the notion of its sum is the Cauchy criterion for the convergence of a series.
		A necessary and sufficient condition for the convergence of the series (5) is that the sequence of its partial sums is bounded above.
	eom.springer.de /s/s084670.htm (2297 words)

	SERIES (a Latin word f... - Online Information article about SERIES (a Latin word f...
		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)

	PlanetMath: convergent series
		which converges, but which is not absolutely convergent is called conditionally convergent.
		It can be shown that absolute convergence implies convergence.
		This is version 7 of convergent series, born on 2002-02-20, modified 2006-11-07.
	planetmath.org /encyclopedia/AbsoluteConvergence.html (178 words)

	Convergent Series -- from Wolfram MathWorld
		A series is said to be convergent if it approaches some limit (D'Angelo and West 2000, p.
		Conversely, a series is divergent if the sequence of partial sums is divergent.
		Convergence and divergence are unaffected by deleting a finite number of terms from the beginning of a series.
	mathworld.wolfram.com /ConvergentSeries.html (151 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.
		The latter two questions are completely answered by Riemann's theorem for rearrangements of arbitrary conditionally convergent series, but our goal is to provide a more concrete setting of Riemann's results within the context of the alternating harmonic series with the hope that the reader will then have a better understanding of the general theory.
	ecademy.agnesscott.edu /~lriddle/series/rear.htm (552 words)

	Series -- from Wolfram MathWorld
		The term infinite series is sometimes used to emphasize the fact that series contain an infinite number of terms.
		Riemann series theorem states that, by a suitable rearrangement of terms, a so-called conditionally convergent series may be made to converge to any desired value, or to diverge.
		An especially strong type of convergence is called uniform convergence, and series which are uniformly convergent have particularly "nice" properties.
	mathworld.wolfram.com /Series.html (793 words)

	Aristotle and Mathematics > The Infinite (Stanford Encyclopedia of Philosophy)
		I shall mean by a convergent series, one that converges on a finite, positive value, and by a non-convergent series, one that increases ad infinitum.
		When we combine the two notion of an infinite series by addition or by division with the notions of a potential or actual series to construct four notions of the infinite.
		The infinite series in potentiality by addition is identical with some series of the infinite in potential by division.
	plato.stanford.edu /entries/aristotle-mathematics/supplement3.html (1231 words)

	PlanetMath: manipulating convergent series
		The terms of the series in the following theorems are supposed to be either real or complex numbers.
		Since all the partial sums of (4) are simultaneously partial sums of (3), they have as limit the sum of the series (3).
		This is version 9 of manipulating convergent series, born on 2004-11-23, modified 2006-10-14.
	planetmath.org /encyclopedia/ManipulatingConvergentSeries.html (142 words)

	Convergent Series (via CobWeb/3.1 planet03.csc.ncsu.edu) (Site not responding. Last check: 2007-10-11)
		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.cob-web.org:8888 /~igc/tch/ma2001/notes/node49.html (262 words)

	Infinity
		It is clear that any series whose terms are equal to or smaller than the terms of a geometric series is a convergent series.
		The sum of an absolutely convergent series can be considered as the sum of the positive terms less the sum of the negative terms with changed sign.
		This series converges too slowly to be a good means of calculating π, but it can be modified to be applicable to this calculation.
	www.du.edu /~etuttle/math/series.htm (2997 words)

	SparkNotes: Series: Terms
		Similarly, a series with positive terms diverges if there is another series with all terms lesser or equal which diverges.
		Convergent - The property that the partial sums of a series have a well-defined limit.
		Series - A sum of the elements in a sequence.
	www.sparknotes.com /math/calcbc2/series/terms.html (357 words)

	Complex Sequences and Series
		The first finitely many terms of a series do not affect its convergence or divergence and, in this respect, the beginning index of a series is irrelevant.
		Hence, Theorem 4.4 implies that the given complex series is convergent.
		, and Theorem 4.5 implies that the series is not convergent; hence it is divergent.
	math.fullerton.edu /mathews/c2003/ComplexSequenceSeriesMod.html (692 words)

	Mean Partial Sums of Non-Convergent Series (Site not responding. Last check: 2007-10-11)
		If the partial sums of an infinite series converge on a finite value the series is said to be convergent, whereas if the partial sums increase (in magnitude) without limit, the series is said to be divergent.
		In addition to these two kinds of series, there is another category of series - which may be called non-convergent - whose partial sums are bounded in magnitude and yet do not converge on any finite value.
		= 1/(1-x) with x=-1 (where the series is non-convergent).
	www.mathpages.com /home/kmath499.htm (417 words)

	Forward Shifts and Backward Shifts in a Rearrangement of a Conditionally Convergent Series American Mathematical ... (Site not responding. Last check: 2007-10-11)
		57] it is proved that a rearrangement of a conditionally convergent series remains convergent (with unaltered sum), provided the series is rearranged in such a way that the forward shifts are bounded.
		The nth term, x^sub n^ = x^sub π(π^sup -1^(n))^, of the original series is shifted to the kth term of the rearranged series, where k = π^sup -1^(n).
		(2) The two series Σ x^sub n^ and Σ x^sub π(n)^ are either both convergent or both divergent; in case of convergence, the sums are identical; for series of real numbers, in case of divergence, either both have an infinite sum or neither does.
	www.findarticles.com /p/articles/mi_qa3742/is_200412/ai_n9520598 (857 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 illustrates the series >converges, but as you say not absolutely, since then it would be the >harmonic series to which you referred.
		It is conditionally convergent, however; use Abel's technique (I think) of showing the partial sums of ((-1)^n)(sin n) are bounded and the 1/n approach zero.
		Abel's Test: Let u(n) be terms of a convergent series, and let v(n) be terms of a monotone convergent sequence.
	www.math.niu.edu /~rusin/known-math/99/series (539 words)

	15.1 Infinite Series and Convergence (Site not responding. Last check: 2007-10-11)
		If we have an infinite sequence, we define it to be convergent if, for any positive criterion, q, however small, beyond some term, say the n(q)th, all of the terms are within q of some number, z which we call the limit of the sequence.
		Then convergence of the series is defined to be the same as the convergence of that sequence of partial sums.
		When a series is absolutely convergent, you can rearrange its terms, differentiate it term by term if terms contain a variable, and perform other manipulations, which may not work for merely convergent series.
	www-math.mit.edu /~djk/calculus_beginners/chapter15/section01.html (292 words)

	MA 109 College Algebra Chapter 5 (Site not responding. Last check: 2007-10-11)
		A series is said to be convergent or conditionally convergent if it converges to some number; otherwise, it is said to be divergent.
		It is important to distinguish between the sequence of terms of an infinite series and the sequence of partial sums of an infinite series.
		For this assertion, it is enough to treat the case where the series consists of non-negative terms.
	www.msc.uky.edu /ken/ma109/lectures/elem.htm (2040 words)

	SparkNotes: Sequences and Series: Terms and Formulae
		Convergent Series - A series whose limit as n→∞ is a real number.
		Divergent Series - A series whose limit as n→∞ is either ∞ or - ∞.
		Finite Series - A series which is defined only for positive integers less than or equal to a certain given integer.
	www.sparknotes.com /math/precalc/sequencesandseries/terms.html (388 words)

	11.4 Absolute Convergence (Site not responding. Last check: 2007-10-11)
		Defining sine and cosine in terms of infinite series can be dangerous to the well being of the definer.
		In 1933 Edmund Landau was forced to resign from his position at the University of Göttingen as a result of a Nazi-organized boycott of his lectures.
		The harmonic series was shown to be unbounded by Nicole
	www.reed.edu /~mayer/html2/node31.html (380 words)

	No Title
		Given a convergent series, we next want to know whether the series is absolutely or conditionally convergent.
		If a series converges, determine, if possible, the value that it converges to.
		Because the series converges conditionally, we can not determine a unique value that it converges to.
	www-math.cudenver.edu /~rrosterm/quizz6sol/quizz6sol.html (215 words)

	Marathon's Story... Facts and puzzling things about
		Niven's "Convergent Series" (published in 1979 by Ballantine Books) is in fact a series of short stories, one of which is called "Convergent Series".
		Centerpiece of Sheffield's Heritage Universe series are the "artifacts", enigmatic structures, seemingly defying natural laws, left by an ancient vanished race.
		Regarding the Convergent Series company, it makes me think that it is called that because Jason Jones converged his efforts to work with Alexander Seropian.
	www.marathon.org /story/convergentseries.html (2324 words)

	Absolute and Conditional Convergence
		This gives one way of proving that a series is convergent even if the terms are not all positive, and so we can't use the comparison test directly.
		We give the proof because the argument is so like the proof of the convergence of the ratio of adjacent terms in the Fibonacci series 3.1.
		Warning:It is not useful to re-arrange conditionally convergent series (remember the rearrangement I did in section 1.1).
	www.maths.abdn.ac.uk /~igc/tch/ma2001/notes/node51.html (496 words)

	Sum of an infinite series
		Afterwards, proceed with a construction of a series of circles on top of each other, each touching the original two circles and its immediate predecessor.
		is a convergent series if ever there was one.
		For this reason, the series in (3) is known as telescoping.
	www.cut-the-knot.org /pythagoras/a_series.shtml (256 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)

	Convergent series - Wikipedia, the free encyclopedia
		If r = 1, the root test is inconclusive, and the series may converge or diverge.
		Also known as the Leibniz criterion, the alternating series test states that for an alternating series of the form
		converges if and only if the sequence of partial sums is a Cauchy sequence.
	en.wikipedia.org /wiki/Convergent_series (435 words)

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