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Topic: Power series

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	Power series - Wikipedia, the free encyclopedia
		These power series arise primarily in analysis, but also occur in combinatorics (under the name of generating functions) and in electrical engineering (under the name of the Z-transform).
		Once a function is given as a power series, it is continuous wherever it converges and is differentiable on the interior of this set.
		Every power series with a positive radius of convergence is analytic on the interior of its region of convergence.
	en.wikipedia.org /wiki/Power_series (1084 words)

	Formal power series - Wikipedia, the free encyclopedia
		However, as a formal power series, we may ignore this completely; all that is relevant is the sequence of coefficients [1, −3, 5, −7, 9, −11,...].
		In a sense, all formal power series are Taylor series.
		It is a topological field, and its relationship to formal power series is analogous to the relationship between power series and Laurent series.
	en.wikipedia.org /wiki/Formal_power_series (2326 words)

	[No title]
		This is the harmonic series, and this series diverges.
		This is the alternating harmonic series, and this series converges.
		The main use of a power series is to numerically approximate integrals or derivative of functions that are not easily integrated or differentiated.
	faculty.eicc.edu /bwood/ma155supplemental/supplemental26.htm (790 words)

	PlanetMath: power series (Site not responding. Last check: 2007-11-07)
		A power series is a series of the form
		Where it converges the power series defines a function, which can thus be represented by a power series.
		This is version 16 of power series, born on 2002-03-19, modified 2005-07-04.
	planetmath.org /encyclopedia/PowerSeries.html (208 words)

	[No title] (Site not responding. Last check: 2007-11-07)
		In mathematics, a power series is an infinite series of the form
		A power series will converge for some values of the variable x (at least for x = a) and may diverge for others.
		This leads to the concept of formal power series, a principle that is of great utility in combinatorics.
	www.informationgenius.com /encyclopedia/p/po/power_series.html (517 words)

	PlanetMath: formal power series (Site not responding. Last check: 2007-11-07)
		Formal power series allow one to employ much of the analytical machinery of power series in settings which don't have natural notions of convergence.
		This latter series is guaranteed to converge in
		This is version 8 of formal power series, born on 2002-07-01, modified 2005-02-13.
	planetmath.org /encyclopedia/FormalPowerSeries.html (762 words)

	Power Series
		Power series are infinitely differentiable functions inside their disk of convergence.
		To obtain the derivative of a power series, one may differentiate it term by term.
		On each compact subset of that disk, the derived series converges uniformly and absolutely to the derivative of the function to which the power series converges.
	www.math.sc.edu /~sharpley/math555/Lectures/Series_Power.html (285 words)

	Power series - maths online Gallery
		Thus it is illustrated how the individual powers contribute to the sum.
		On demand, some examples of (truncated) power series of given functions are displayed.
		Hence, it is better suited for the detailed analysis of particular power series.
	www.univie.ac.at /future.media/moe/galerie/potr/potr.html (195 words)

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	IBM POWER Encyclopedia Article, Definition, History, Biography (Site not responding. Last check: 2007-11-07)
		The POWER series microprocessors are used as the main CPU in many of IBM's servers, minicomputers, workstations, and supercomputers.
		The POWER architecture was used to develop (and remains very similar to) the PowerPC architecture, used in later Apple Macintosh computers, some IBM workstations, as well as a number of embedded applications.
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	encyclopedia.localcolorart.com /encyclopedia/IBM_POWER (1254 words)

	Taylor Series
		You should also be familiar with the geometric series, the notion of a power series, and in particular the concept of the radius of convergence of a power series.
		This is where the notion of the radius of convergence of a power series will become useful.
		In an earlier example (the example is almost identical!), we saw that this power series has a radius of convergence of 1.
	www.sosmath.com /calculus/tayser/tayser01/tayser01.html (539 words)

	Power Series (Site not responding. Last check: 2007-11-07)
		ANTIOPIC The Allegorical Power Series Volume III August 2003...
		Volume 24 "Formal Power Series and Algebraic Combinatorics" Louis J. Billera, Cu...
		Bilinear operators and the power series for the...
	www.scienceoxygen.com /math/224.html (143 words)

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	www.kepcopower.com (1150 words)

	Power Series Applications (Site not responding. Last check: 2007-11-07)
		Rich Robinson ==================================================== Power series are infinite series that contain terms X^r, that is X raised to a series of exponents.
		There are some very elegant infinite series such as the power series expression called Fibonacci series (you can find a lot of books on these).
		There are special groups of power series such as Taylor series that find wide use in many applications.
	www.newton.dep.anl.gov /askasci/math99/math99165.htm (469 words)

	Women of Power Series
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	www.mith.umd.edu /flare/womenpower (311 words)

	15.5 Power Series
		Power series supply us with a new way to describe functions: we can specify the coefficients of each power in the series.
		We can also look at power series for which we subtract some value, call it z, from the variable x.
		Since series converge when this ratio is any factor fixed factor less than 1, power series typically converge up to some maximum magnitude of the expansion variable, which value is the limiting ratio of
	www-math.mit.edu /~djk/calculus_beginners/chapter15/section05.html (450 words)

	Introduction to power series (Site not responding. Last check: 2007-11-07)
		A power series is a series of increasing powers of x, with each term multiplied by a number (called a coefficient).
		The number of the subscript is always the same as the power of x that the coefficient multiplies.
		Other functions may also be expressed as power series, as we'll see in later sections.
	www.ucl.ac.uk /Mathematics/geomath/level2/series/ser7.html (150 words)

	Other Power Series
		We now derive some further power series to add to the collection described in 7.1.
		Apply the ratio test to the given power series.
		We have now been able to derive a power series representation for a function without working directly from the Taylor series, and doing the differentiations -- which can often prove very awkward.
	www.maths.abdn.ac.uk /~igc/tch/ma2001/notes/node56.html (356 words)

	power series (Site not responding. Last check: 2007-11-07)
		The theorem of Taylor predicts that a function that behaves sufficient neatly can be written as a power series of polynomials.
		This series can be used to define a function (with curve y = f(x)).
		In that way trigonometric functions can be built, without using geometrical insights.
	www.2dcurves.com /power/powers.html (70 words)

	Muscle Power Series List
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	Power series (Site not responding. Last check: 2007-11-07)
		The series fails to converge on and beyond the unit circle, as is clear from the series for
		This expansion converges in the right-hand circle, which overlaps with the unit circle in the shaded region.
		This is called analytic continuation, and in this way one can cover the whole of the complex plane, with the exception of singularities, by convergent Taylor series expansions of the function.
	www.astro.cf.ac.uk /undergrad/module/PX3211/com/node4.html (273 words)

	Math Forum - Ask Dr. Math (Site not responding. Last check: 2007-11-07)
		Date: 08/31/98 at 20:44:51 From: Sam Subject: Long division to find power series Can you show me how to find a power series to approximate this function: 1/(1-x)^2 using long division.
		The power series for that function is 1 + 2x + 3x^2 + 4x^3 +....
		Date: 09/01/98 at 02:18:05 From: Doctor Floor Subject: Re: Long division to find power series Thank you for sending in your question.
	mathforum.org /dr.math/problems/sam8.31.98.html (200 words)

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	Preschool Activity Children's Videos: The Preschool Power! Series
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	www.preschoolpower.com (470 words)

	Power Series
		and use the formula for the geometric series with
		With a little bit of work, the formula for the geometric series has led to a series expression for the inverse tangent function!
		Such expressions are called power series with center 0; the numbers
	www.sosmath.com /calculus/powser/powser01.html (228 words)

	Visual Calculus - Power Series - 2 (Site not responding. Last check: 2007-11-07)
		Suppose that the radius of convergence of the power series
		In addition, the radius of convergence of these two power series is also r.
		Find the derivative and the integral of the function
	archives.math.utk.edu /visual.calculus/6/power.2 (47 words)

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