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Topic: Maclaurin series

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	Karl's Calculus Tutor - Little Red Riding Hood Goes to Town (approximation & intro to Tayor & Maclaurin ...
		The radius of convergence for this Maclaurin series is
		There is much more going on than meets the eye with Maclaurin series, and you will learn a lot more about it when you learn about how the domains of ordinary functions can be expanded to include complex numbers.
		If learning to find a Maclaurin series were learning to tie your shoes when you are at home, then finding a Taylor series is no more different than tying your shoes at school.
	www.karlscalculus.org /calc8_3.html (4669 words)

	Maclaurin Series
		The subject of Maclaurin Series is one that will be carefully justified in Calculus.
		The series can be shifted to any other number if the required input is not near zero.
		In this case, the series are known as Taylor Series.
	mathcircle.berkeley.edu /BMC4/Handouts/serie/node4.html (219 words)

	S.O.S. Mathematics CyberBoard :: View topic - Maclaurin series
		Posted: Wed, 14 Dec 2005 23:05:17 GMT Post subject: Maclaurin series
		Series of one times the series of the other, I believe.
		Posted: Thu, 15 Dec 2005 01:09:45 GMT Post subject: Re: Maclaurin series
	www.sosmath.com /CBB/viewtopic.php?t=19970&start=0&postdays=0&postorder=asc&highlight= (174 words)

	A Gallery of Complex Functions (page 2)
		The function f(z) = exp(z) is special because it has no zeros or poles on the complex plane, yet it is not a constant function.
		The only possibility is that when a term of the series is added it adds one more zero, but it also slightly push every zero away.
		At the limit, no zero is left because they all have been pushed to infinity.
	www.cs.berkeley.edu /~flab/complex/gallery2.html (113 words)

	PlanetMath: Taylor series
		That the series on the right converge to the functions on the left can be proven by Taylor's Theorem.
		Taylor series and polynomials can be generalized to Banach spaces: for details, see Taylor's formula in Banach spaces.
		This is version 20 of Taylor series, born on 2001-11-08, modified 2006-12-27.
	planetmath.org /encyclopedia/TaylorSeries.html (798 words)

	Colin Maclaurin (1698 - 1746)
		Maclaurin took an active part in opposing the advance of the Young Pretender in 1745; on the approach of the Highlanders he fled to York, but the exposure in the trenches at Edinburgh and the privations he endured in his escape proved fatal to him.
		Maclaurin also shewed that a spheroid was a possible form of equilibrium of a mass of homogeneous liquid rotating about an axis passing through its centre of mass.
		Maclaurin was one of the most able mathematicians of the eighteenth century, but his influence on the progress of British mathematics was on the whole unfortunate.
	www.maths.tcd.ie /pub/HistMath/People/Maclaurin/RouseBall/RB_Maclaurin.html (1049 words)

	Colin Maclaurin Summary
		Maclaurin was born in Kilmodan, Scotland, in 1698, the son of a minister named John Maclaurin, a man of great learning.
		Maclaurin, for instance, showed that the cubic and the quartic could be represented by rotating these angles around their vertices.
		Political events soon intervened in Maclaurin's life, as in 1745 a revolt broke out among the Jacobites, a faction who claimed that the Catholic line descended from James II, second son of Charles I, were the rightful heirs to the British throne rather than the descendants of James's elder brother Charles II.
	www.bookrags.com /Colin_Maclaurin (1794 words)

	Maclaurin biography
		However, Maclaurin had to defend a thesis in a public examination for the award of this degree (which is not the case today), and he chose On the power of gravity as his topic.
		Maclaurin had already shown himself a very strong advocate of the mathematical and physical ideas of Newton, so it was natural that they should meet during Maclaurin's visit to London.
		Maclaurin himself acted as one of the two secretaries of this expanded Society and at the monthly meetings he often read a paper of his own or a letter from a foreign scientist on the latest developments in some topic of current interest.
	www-groups.dcs.st-and.ac.uk /~history/Biographies/Maclaurin.html (2510 words)

	Taylor series Summary
		A Taylor's series is a series expansion that acts as a representation of a function.
		He found a number of special cases of the Taylor series, including the Taylor series for the trigonometric functions of sine, cosine, tangent and arctangent, and the second-order Taylor series approximations of the sine and cosine functions, which he extended to the third-order Taylor series approximation of the sine function.
		The Maclaurin series was named after Colin Maclaurin, a professor in Edinburgh, who published the special case of the Taylor result in the 17th century.
	www.bookrags.com /Taylor_series (1855 words)

	Series (Site not responding. Last check: )
		For instance in the case, f(x)= 1/x the function is not defined at x=0, and therefore the series expansion fails to reproduce the behaviour of the function in that limit.
		This example shows that the series expansion of the function is valid only in the region about the point chosen for the expansion.
		Series expansions are just polynomial functions, therefore one should take this into account when fitting data to this type of functionality.
	metric.ma.ic.ac.uk /series/series_power/series_power_taylor.html (215 words)

	Series (Site not responding. Last check: )
		Series expansions rely on the concept of derivative.
		One important issue in approximating a function through a series expansions, is the number of terms we need to include to represent the function with certain accuracy.
		In principle we can approach the function using any number of terms in the series, but unless we are considering an infinite number of those terms, there will be always a difference between the original function and the expansion.
	metric.ma.ic.ac.uk /series/series_power/series_power_expansion.html (513 words)

	Project Links \| Drag Forces Module
		Again, the preceding series may be shown to converge for all values of.
		In the term of the series the last factor of the numerator is.
		It is tempting to multiply the power series like polynomials, that is, assume that the series for the product of two functions is equal to the product of the series, then carry out term-by-term multiplication of the two series.
	links.math.rpi.edu /devmodules/dragforces/html/taylor_series.html (571 words)

	Doglas, Y. (2006). Pseudo- Maclaurin Expansion. PHILICA.COM Article number 6.
		Furthermore, in simple functions such as x^2, the sum of the series is an algebraic progression, which can be greatly simplified.
		In contrast to the Maclaurin expansion, all n derivatives would have to be calculated to find f(x+nh).
		This theorem is no doubt less useful than Maclaurin’s theorem, but it could be useful in cases whereby only the first derivative is known.
	philica.com /display_article.php?article_id=6 (930 words)

	CALCULUS II EXAM III NOTES AND LINKS
		You need to be able to recognize a geometric series and be able to sum (determine what it converges to) a convergent infinite geometric series.
		You need to be able to apply the Alternating Series Test to determine convergence of series to which the test applies and to be able to put a bound on the error in summing the first n terms of a convergent alternating series by using the Alternating Series Remainder Theorem (P 633).
		You need to be familiar with the power series representations for elementary functions shown on page 682 of your text and be able to use these results to construct power series representations of other functions.
	www2.scc-fl.edu /lvosbury/CalculusII_Folder/Calculus_II_Exam_3.htm (1809 words)

	Maclaurin (Site not responding. Last check: )
		Maclaurin was a child prodigy who began his studies at the University of Glasgow at age eleven; at age fourteen he received his master's degree.
		Maclaurin is really one of the big early contributors to calculus.
		Nowadays Maclaurin is best known for something he never claimed to have invented, the Maclaurin series which is just the Taylor series centered at the origin.
	www.math.fau.edu /schonbek/Modern_Analysis/calcmath12.html (382 words)

	powerseriesI.htm
		A second motivation for the use of power series techniques lies in the realm of partial differential equations and a fundamental method for solving them, separation of variables.
		The bottom line here is that power series techniques give a powerful approach to the study of solutions of Bessel's equation and many others arising in mathematical physics.
		A fact from calculus comes to the rescue here: power series may be differentiated term by term in their interval of convergence and the resulting series has the same interval of convergence and is the derivative of the original series.
	germain.umemat.maine.edu /faculty/bray/Archive_notes/powerseriesI.htm (1293 words)

	[No title]
		For any Taylor or Maclaurin series, we can use the method we just used to determine the interval of convergence of a power series.
		Here are some specific Maclaurin series and their interval of convergence that you should know.
		There are more Maclaurin series stated in the textbook, but these are the ones that are used most often.
	faculty.eicc.edu /bwood/ma155supplemental/supplemental28.htm (459 words)

	Taylor's series (Site not responding. Last check: )
		We've seen in the last few sections that it can be useful to write functions as infinite series of powers of x, in particular this allows us to find approximate values of the functions for different values of x.
		For the series of f(x), we're not going to write a series of powers of x, instead a series of powers of (x-a).
		MacLaurin's series is the same as Taylor's series but with a=0.
	www.ucl.ac.uk /Mathematics/geomath/level2/series/ser12.html (524 words)

	Taylor expansions
		Such an approximation is known by various names: Taylor expansion, Taylor polynomial, finite Taylor series, truncated Taylor series, asymptotic expansion, Nth-order approximation, or (when f is defined by an algebraic or differential equation instead of an explicit formula) a solution by perturbation theory (see below).
		This is an elaboration of Exercise 30 of Sec.
		Now that we have presented the basic ideas about finite Taylor series and their applications, you may be ready to give Sections 10.9-12 a first reading.
	www.math.tamu.edu /~fulling/coalweb/taylor.htm (1811 words)

	Laurent Series Representations
		In this case, therefore, there are no negative powers involved, and the Laurent series reduces to the Taylor series.
		The uniqueness of the Laurent series is an important property because the coefficients in the Laurent expansion of a function are seldom found by using Equation
		which is a Laurent series that reduces to a Maclaurin series.
	math.fullerton.edu /mathews/c2003/LaurentSeriesMod.html (332 words)

	TaylorDemo.html
		It also extends the fundamentals of series concepts to new areas of mathematical studies.
		There is no elementary antiderivative for the integrand function but we can create an infinite power series representation for this integral and then integrate term-by-term to obtain a series solution for the antiderivative.
		We can use a series to approximate the value of a difficult integral to a desired accuracy.
	www.mc.maricopa.edu /~dschultz/TaylorDemo.html (896 words)

	Taylor Polynomials and their Graphs (Site not responding. Last check: )
		These polynomial approximations are the well-known Taylor-Maclaurin series approximations, for which one needs only the value of the function and of its derivatives at a single specified point.
		These polynomials lead to infinite series representations of the function (on the interval of convergence) that may then be manipulated algebraically, differentiated, integrated, and so on.
		Creating series approximations for any transcendental functions that you define by specifying recursive relations on the derivative and the definition at a single evaluation point.
	www.mathwright.com /book_pgs/book263.html (248 words)

	The Maclaurin series generated by f(x)=x^3·cosx + 1 (Site not responding. Last check: )
		+ ··· for all x in some interval containing a, then the series is the Taylor series for f about a.
		What this means is that if you can construct a series representation for the function you have that converges in an interval containing 0, then it is the Maclaurin series.
		For you example take the Maclaurin series representation of cos(x), multiply each term by x
	mathcentral.uregina.ca /QQ/database/QQ.09.04/latto1.html (104 words)

	notes4_1
		Given this, once we have memorized a few basic power series and know their radius of convergence we can easily find more complicated series and easily know their radius of convergence without having to perform the ratio test.
		Lets look at a few partial sums to see that we have some ``good'' polynomial approximations of the functions we are working with (as long as we are within the radius of convergence).
		What is missing is that we do not know the entire domain of these new series; i.e., we do not know if these series converge or diverge at the endpoints of the intervals of convergence.
	www-math.cudenver.edu /~rrosterm/notes4_1/notes4_1.html (435 words)

	Taylor Series Section 4. Examples
		, the truncation of the Taylor series at the
		Its most obvious connection to this area of mathematics is the geometric series.
		As a matter of practice, I leave it to the reader to actually “compute” this series using the derivatives and the definition of Maclaurin series.
	www.krellinst.org /UCES/archive/resources/taylor_series/node4.shtml (637 words)

	Maclaurin, Colin (1698-1746)
		A Scottish mathematician who developed and extended Isaac Newton's work on calculus and gravitation, and did notable work on higher plane curves.
		In his Treatise of Fluxions (1742), he gave the first systematic formulation of Newton's methods and set out a method for expanding functions about the origin in terms of series now known as Maclaurin series.
		Maclaurin also invented several devices, made astronomical observations, wrote on the structure of bees' honeycombs, and improved maps of the Scottish isles.
	www.daviddarling.info /encyclopedia/M/Maclaurin.html (149 words)

	[No title]
		The above series is called the Taylor series generated by f at x = a.
		The Taylor series is an infinite series, whereas a Taylor polynomial is a polynomial of degree n and has a finite number of terms.
		Now that I have introduced the topic of power, Taylor, and Maclaurin series, we will now be ready to determine Taylor or Maclaurin series for specific functions.
	faculty.eicc.edu /bwood/ma155supplemental/supplemental27.htm (563 words)

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