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Topic: Taylor series

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Taylor expansion

Laurent series

Power series

Even and odd functions

Holomorphic function

Exponential function

Hyperbolic function

Natural logarithm

Squeeze theorem

Fourier series

Residue (complex analysis)

Trigonometric function

Geometric series

Riemann zeta function

Intermediate value theorem

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	Taylor series - Wikipedia, the free encyclopedia
		The Taylor series, power series, and infinite series expansions of functions may have been first discovered in India by Madhava in the 14th century.
		He is also thought to have discovered the power series of the radius, diameter, circumference, angle θ, π and π/4, along with rational approximations of π, and infinite continued fractions.
		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.
	en.wikipedia.org /wiki/Taylor_series (1202 words)

	PlanetMath: Taylor series
		In contrast with the complex case, it turns out that all holomorphic functions are infinitely differentiable and have Taylor series that converge to them.
		Taylor series and polynomials can be generalized to Banach spaces: for details, see Taylor's formula in Banach spaces.
		This is version 19 of Taylor series, born on 2001-11-08, modified 2005-10-14.
	planetmath.org /encyclopedia/TaylorSeries.html (745 words)

	Taylor biography
		Taylor was brought up in a household where his father ruled as a strict disciplinarian, yet he was a man of culture with interests in painting and music.
		Taylor initially derived the version which occurs as Proposition 11 as a generalisation of Halley's method of approximating roots of the Kepler equation, but soon discovered that it was a consequence of the Bernoulli series.
		Taylor, in his studies of vibrating strings was not attempting to establish equations of motion, but was considering the oscillation of a flexible string in terms of the isochrony of the pendulum.
	www-groups.dcs.st-and.ac.uk /~history/Biographies/Taylor.html (1927 words)

	PlanetMath: Taylor series of arcus sine
		We give an example of obtaining the Taylor series expansion of an elementary function by integrating the Taylor series of its derivative.
		"Taylor series of arcus sine" is owned by pahio.
		This is version 8 of Taylor series of arcus sine, born on 2004-11-24, modified 2006-02-22.
	planetmath.org /encyclopedia/TaylorSeriesOfArcusSine.html (171 words)

	Taylor series
		Limited sets of infinite series expansions for particular functions were known in India by Madhava in the fourteenth century.
		It was not until 1715 that a general method for constructing these series for all functions for which they exist was finally provided by Brook Taylor, after whom the series are now named.
		If this series converges for every x in the interval (a − r, a + r) and the sum is equal to f(x), then the function f(x) is called analytic.
	www.brainyencyclopedia.com /encyclopedia/t/ta/taylor_series.html (866 words)

	Learn more about Taylor series in the online encyclopedia. (Site not responding. Last check: 2007-09-19)
		If this series converges for every x in the interval (a-r, a+r) and the sum is equal to f(x), then the function f(x) is called analytic.
		is not analytic, the Taylor series is 0, although the function is not.
		The Parker-Sockacki theorem is a recent advance in finding Taylor series which are solutions to differential equations.
	www.onlineencyclopedia.org /t/ta/taylor_series.html (509 words)

	Taylor Series
		Then we require that at c, the power series and the function agree at c; and that all derivatives of the power series equal the corresponding derivatives of F(x).
		The Taylor series for F(x) at c is not necessarily equal to F(x) on the series's interval of convergence.
		In practice the Taylor series does converge to the function for most functions of interest, so that the Taylor series for a function is an excellent way to work that function.
	www.math.wpi.edu /Course_Materials/MA1023C98/taylor/node1.html (1238 words)

	Taylor series (Site not responding. Last check: 2007-09-19)
		By analytic continuation (employing the translation), one may move the point about which the Taylor's series is expanded to any other point within the original circle of convergence.
		In this manner, one may walk the point about which the Taylor's series is expanded to anywhere that the function has all derivatives; provided that a suitable path can be found.
		The specific Taylor's series for the square root is displayed in the binomial.
	www.rism.com /Trig/taylor.htm (223 words)

	Taylor - Wikipedia, the free encyclopedia
		Electoral district of Taylor, a state electoral district in South Australia.
		Taylor, Australian Capital Territory a planned suburb in the Canberra district of Gungahlin
		Taylor rule, in economics, a proposed policy stipulating how interest rate should be changed
	en.wikipedia.org /wiki/Taylor (228 words)

	Elizabeth Taylor Series - Uncyclopedia
		The Elizabeth Taylor series is a collection of DVDs recently release by Warner’s and a mathematical model devised by popular new mathematician Prof.
		Since the Elizabeth Taylor series approximates the truth by adding together lots of nonsense, the sum to infinity of a lot of 'fucking nonsence' is the exact answer.
		The sum of the Elizabeth Taylor series has to be raised by a factor of ten every time a persons IQ is raised by one point.
	www.uncyclopedia.org /wiki/Elizabeth_Taylor_Series (746 words)

	Finding Taylor Series
		This was, in fact, the original use of Taylor series.
		This series is the alternating harmonic series and we have seen that it converges.
		Since this series converges for all values of x, it actually defines a function of x given by evaluating the series.
	www.ugrad.math.ubc.ca /coursedoc/math101/notes/series/taylor.html (695 words)

	Taylor series (Site not responding. Last check: 2007-09-19)
		In mathematics, the Taylor series of an infinitely often differentiable real (or complex) function f defined on an open interval (a-r,a+r) is the power series
		Some functions cannot be written as Taylor series because they have a singularity ; in these cases, one can often still achieve a series expansion if one allows alsonegative powers of the variable x; see Laurent series.
		The Parker-Sockacki theorem is a recent advance in finding Taylor series which aresolutions to differential equations.
	www.therfcc.org /taylor-series-39879.html (445 words)

	Functions and Taylor Series
		The taylor series is the taylor polynomial of degree n, and that polynomial happens to be f.
		This is not surprising; the taylor polynomial ought to equal the polynomial that produced it.
		Its taylor series is 0, yet the function is obviously not 0 for all x.
	www.mathreference.com /ca,tfn.html (514 words)

	Taylor Series
		On their intervals of convergences, Taylor series can be added, subtracted, and multiplied by constants and powers of x, and the resulting series will also be Taylor series.
		To find a series for y = cos 2x, simply substitute 2x in for each x in the Taylor series for cos x.
		Notice, this is not always the same as writing three terms of a series approximating a function.
	www.mecca.org /~halfacre/MATH/taylorseries.htm (230 words)

	Constructing Taylor Series
		In particular, all the derivatives of f at 0 vanish, and the Maclaurin series for f is identically 0.
		A power series may be integrated or differentiated term-by-term in the interior of its interval of convergence.
		I'll use the fact that a Taylor series can be integrated term-by-term on the interval where it converges absolutely.
	marauder.millersville.edu /~bikenaga/calculus/taylor/taylor.html (457 words)

	3. Taylor Series
		Use MATLAB to graphically compare a function with its Taylor polynomial approximations.
		To use Taylor's theorem to estimate the error between f and T_4 we will need the 5'th derivative of f.
		From this picture, we see that absolute value of the 5'th derivative of f is bounded by 25 on the interval of interest.
	math.ucsd.edu /~driver/21d-f99/Taylor_series/Taylor.htm (589 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.
		So you should expect the Taylor series of a function to be found by the same formula as the Taylor polynomials of a function: Given a function f(x) and a center
		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)

	The CTK Exchange Forums
		The taylor series expansion seems to be the same for both g(x+a) and h(x+a) as they depend only on the derivative of f(x) at the point a.
		For one, a Taylor series may have radius of convergence 0.
		The function that is the limit of its Taylor series is analytic, which is a more demanding property than infinite differentiability.
	www.cut-the-knot.org /htdocs/dcforum/DCForumID6/581.shtml (251 words)

	Taylor series
		The successive terms of the Taylor series are decreasing in magnitude in an exponential way.
		This divides the Taylor series in two separate series: the numerical scheme and the error series.
		The error after truncation is mainly dependent on the first term of the error series.
	www.genesis-sim.org /GENESIS/gum-tutorials/cornelis/doc/html/node5.html (101 words)

	Taylor series (Site not responding. Last check: 2007-09-19)
		One simple way to interpret Taylor series is to think of them as a `best fit' polynomial approximation to a function.
		The geometric series is a special case where the error after summing a finite number of terms can be calculated simply.
		Showing the series converges is equivalent, then, to showing that the remainder term goes to zero.
	www.math.columbia.edu /~psorin/calcIIS/taylor/taylor.html (694 words)

	Taylor Series
		A Taylor series is a technique for estimating many important functions to any desired degree of accuracy.
		Power series are a topic in all 1st year Calculus courses; I won't try to explain here how or why they work.
		The "Taylor" who loaned his name to the series referenced here is Brook Taylor, an English mathematician who developed the technique in the early 1700's and whose handsome picture appears at the top of this page.
	www.delphiforfun.org /Programs/Math_Topics/taylor_series.htm (395 words)

	Differential Equations (Math 3401) - Series Solutions to DE's - Review : Taylor Series (Site not responding. Last check: 2007-09-19)
		series once we get out of the review, but they are a nice way to get us back into the swing of dealing with power series. By time most students reach this stage in their mathematical career they’ve not had to deal with power series for at least a semester or two. Remembering how
		series work will be a very convenient way to get comfortable with power series before we start looking at differential equations.
		series we do not multiply the 4 through on the second term or square out the third term. All the terms with the exception of the constant should contain an x-2.
	tutorial.math.lamar.edu /AllBrowsers/3401/TaylorSeries.asp (618 words)

	SparkNotes: The Taylor Series: Approximating Functions With Polynomials
		The theory of Taylor polynomials and Taylor series rests upon once crucial insight: in order to approximate a function, it is often enough to approximate its value and its derivatives (first, second, third, and so on) at one point.
		This series, called the Taylor series of f at 0, is a special kind of power series, an object that was explored in the last chapter.
		In many cases, the Taylor series will define a function that is equal to the original function f (x) inside this radius of convergence.
	www.sparknotes.com /math/calcbc2/taylorseries/section1.html (467 words)

	Taylor Series Section 1. Why Taylor Series?
		Without attempting to delve too far into this question, we give a few examples here of problems that lend themselves nicely to the use of a Taylor series expansion.
		in a certain interval, it is sometimes easier to determine a Taylor series expansion of f, truncate this to a polynomial
		Hence, it is sometimes the case that solutions will be found in terms of Taylor series as the integration is deemed "performable" in this context.
	www.krellinst.org /UCES/archive/resources/taylor_series/node1.shtml (347 words)

	Calculus II (Math 2414) - Series & Sequences - Taylor Series
		Before working any examples of Taylor Series we first need to address the assumption that a Taylor Series will in fact exist for a given function. Let’s first get some notation out of the way first.
		This idea of renumbering the series terms as we did in the previous example isn’t used all that often, but occasionally is very useful. There is one more series where we need to do it so let’s take a look at that so we can get one more example down of renumbering series terms.
		So, we’ve seen quite a few examples of Taylor Series to this point and in all of them we where able to find general formulas for the series. This won’t always be the case. To see an example of one that doesn’t have a general formula check out the last example in the next section.
	tutorial.math.lamar.edu /AllBrowsers/2414/TaylorSeries.asp (1510 words)

	Taylor Series
		Taylor series and orthogonality of the octonion analytic functions.
		A Rudin-Carleson theorem for uniformly convergent Taylor series.
		On a Criterion of Pringsheim's for Expansibility in Taylor's Series
	math.fullerton.edu /mathews/c2003/TaylorSeriesBib/Links/TaylorSeriesBib_lnk_2.html (585 words)

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