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.
As the degree of the Taylor polynomial rises, it approaches the correct function.
^ The degree Taylor polynomial and the degree Taylor polynomial are equal: .

^ Find the third-degree Taylor polynomial for at 0.
  • esm_barnett_appliedcalc_8|Taylor Polynomials and Infinite Series|Quiz 1 20 January 2010 23:45 UTC wps.prenhall.com [Source type: General]

^ Find the degree Taylor polynomial for at .

This image shows sinx (in black) and Taylor approximations, polynomials of degree 1, 3, 5, 7, 9, 11 and 13.
The exponential function (in blue), and the sum of the first n+1 terms of its Taylor series at 0 (in red).
.In mathematics, the Taylor series is a representation of a function as an infinite sum of terms calculated from the values of its derivatives at a single point.^ The Taylor series of is the sum of the Taylor series of and of .
  • Taylor Series. 20 January 2010 23:45 UTC ndp.jct.ac.il [Source type: FILTERED WITH BAYES]

^ For a function of a single variable, , the Taylor Series, , about a point is given by .
  • Taylor Series - Math Images 20 January 2010 23:45 UTC mathforum.org [Source type: FILTERED WITH BAYES]

^ So, provided a power series representation for the function about exists the Taylor Series for about is, .
  • http://tutorial.math.lamar.edu/Classes/CalcII/TaylorSeries.aspx 20 January 2010 23:45 UTC tutorial.math.lamar.edu [Source type: FILTERED WITH BAYES]

.It is named after the English mathematician Brook Taylor.^ The series is named for the English mathematician Brook Taylor .
  • Taylor series (mathematics) -- Britannica Online Encyclopedia 20 January 2010 23:45 UTC www.britannica.com [Source type: General]

^ The Taylor series is named for mathematician Brook Taylor , who first published the power series formula in 1715.
  • Calculus/Taylor series - Wikibooks, collection of open-content textbooks 20 January 2010 23:45 UTC en.wikibooks.org [Source type: Reference]

^ It came to acquire the name Taylor, however, from the 18th century mathematician Brook Taylor.

.If the series is centered at zero, the series is also called a Maclaurin series, named after the Scottish mathematician Colin Maclaurin.^ Find the power series centered at zero (Maclaurin series) representation for .
  • Taylor Polynomials and Taylor Series 20 January 2010 23:45 UTC www2.scc-fl.edu [Source type: Reference]

^ Maclaudian series were named after the Scottish mathematician Colin Maclaudin.

^ If a = 0 the series is called a Maclaurin series, after the Scottish mathematician Colin Maclaurin .
  • Taylor series (mathematics) -- Britannica Online Encyclopedia 20 January 2010 23:45 UTC www.britannica.com [Source type: General]

.It is common practice to use a finite number of terms of the series to approximate a function.^ Maclaurin series will be useful for when the function is being approximated for small values of x.

^ Approximate using the first two nonzero terms of the binomial series.

^ Maclaurin series for common functions include .
  • Maclaurin Series -- from Wolfram MathWorld 20 January 2010 23:45 UTC mathworld.wolfram.com [Source type: General]

.The Taylor series may be regarded as the limit of the Taylor polynomials.^ Taylor 700 Series Limited Madagascar b   .
  • Taylor 700 Series Limited Madagascar b 20 January 2010 23:45 UTC www.12fret.com [Source type: General]

^ The Taylor series for a polynomial is just the polynomial itself.
  • Taylor Series and Maclaurin Series, AP Calculus II/BC - Educator.com 20 January 2010 23:45 UTC www.educator.com [Source type: General]

^ Taylor series as limits of Taylor polynomials .
  • Taylor Series 20 January 2010 23:45 UTC www.sosmath.com [Source type: General]

Contents

Definition

The Taylor series of a real or complex function ƒ(x) that is infinitely differentiable in a neighborhood of a real or complex number a is the power series
f(a)+\frac {f'(a)}{1!} (x-a)+ \frac{f''(a)}{2!} (x-a)^2+\frac{f^{(3)}(a)}{3!}(x-a)^3+ \cdots
which can be written in the more compact sigma notation as
 \sum_{n=0} ^ {\infin } \frac {f^{(n)}(a)}{n!} \, (x-a)^{n},
where n! denotes the .factorial of n and ƒ (n)(a) denotes the nth derivative of ƒ evaluated at the point a.^ A Taylor series provides a way to generate such a series and is computed as: where f is the function for which we want a series representation and is the n th derivative of f evaluated at c .
  • Taylor Series and Polynomials 20 January 2010 23:45 UTC www.calculusapplets.com [Source type: FILTERED WITH BAYES]
  • Taylor Series and Polynomials — Notre Dame OpenCourseWare 20 January 2010 23:45 UTC ocw.nd.edu [Source type: Reference]

^ Thus the explicit value for the remainder in the integral form, which requires knowing the function f ( x ) well enough to be able to evaluate the integral, has been replaced by an form dependent only on the ( n +1) th derivative of f ( x ), but at some point, , unknown to us, except that it lies between the values x and a .

^ The nth coefficient is just the nth derivative of the original function, evaluated at c, divided by n factorial.
  • The Idea Shop: An Easy Way to Remember the Taylor Series Expansion 20 January 2010 23:45 UTC www.the-idea-shop.com [Source type: FILTERED WITH BAYES]

The zeroth derivative of ƒ is defined to be ƒ itself and (xa)0 and 0! are both defined to be 1.

Maclaurin series

In the particular case where a = 0, the series is also called a Maclaurin series:
f(0)+f'(0)x + \frac{f''(0)}{2!}x^2+\frac{f^{(3)}(0)}{3!}x^3+ \cdots

Derivation

.The Maclaurin / Taylor series can be derived in the following manner.^ How to Derive a Taylor Series • vegetarianism .

^ The following equation is the defining equation of a Taylor series: .

^ MacLaurin's series is the same as Taylor's series but with a=0.
  • Taylor's series 20 January 2010 23:45 UTC www.ucl.ac.uk [Source type: FILTERED WITH BAYES]

.An arbitrary function may be defined by a power series:
 f(x) = \sum_{n=0}^\infty a_n x^n = a_0 + a_1 x + a_2 x^2 + a_3 x^3 + \ldots.</span><span class=^ Many functions can be written as a power series.
  • Taylor Series 20 January 2010 23:45 UTC www.sosmath.com [Source type: General]

^ The function can be expanded in a power series in power of as .

^ The power series in x – a for a function f .
  • Mathwords: Taylor Series 20 January 2010 23:45 UTC www.mathwords.com [Source type: Academic]

" src="http://images-mediawiki-sites.thefullwiki.org/05/4/0/4/029829578480820.png" />

Evaluating at x = 0, we have:
f(0) = a0

Differentiating the function,
f'(x) = a_1 + 2 a_2 x + 3 a_3 x^2 + 4 a_4 x^3 + \cdots

Evaluating at x = 0,
f'(0) = a1

Differentiating the function again,
f''(x) = 2 a_2 + 6 a_3 x + 12 a_4 x^2 + \cdots

Evaluating at x = 0,
\frac{f''(0)}{2!} = a_2

Generalizing,
a_n = \frac{f^n(0)}{n!} Where fn(0) is the nth derivative of f(0).

.Substituting the respective values of an in the power expansion,
f(x) = f(0) + f'(0) x + \frac{f''(0)}{2!} x^2 + \frac{f'''(0)}{3!} x^3 + \cdots
Which is a particular case of the Taylor series (also known as Maclaurin series).
^ In the case of a=0, the series expansion is called a Maclaurin series .
  • Taylor Series - Math Images 20 January 2010 23:45 UTC mathforum.org [Source type: FILTERED WITH BAYES]

^ Then the Taylor expansion is called a Maclaurin expansion .
  • Taylor expansions 20 January 2010 23:45 UTC www.math.tamu.edu [Source type: FILTERED WITH BAYES]

^ Taylor expansions: The syntax for computing Taylor series is: .
  • TI-89 tutorial 19 September 2009 13:16 UTC pages.infinit.net [Source type: Reference]



.Generalizing further, we have
f(a + x) = f(a) + f'(a) x + \frac{f''(a)}{2!} x^2 + \frac{f'''(a)}{3!} x^3 + \cdots
Which is the Taylor series.
^ The general formula for a Taylor Series centered at x = a is: .
  • Taylor Series@Everything2.com 20 January 2010 23:45 UTC www.everything2.com [Source type: General]

^ General principles on accuracy of Taylor Series: .
  • Taylor Polynomial Applications, AP Calculus II/BC - Educator.com 20 January 2010 23:45 UTC www.educator.com [Source type: Reference]

^ A Taylor series provides a way to generate such a series and is computed as: where f is the function for which we want a series representation and is the n th derivative of f evaluated at c .
  • Taylor Series and Polynomials — Notre Dame OpenCourseWare 20 January 2010 23:45 UTC ocw.nd.edu [Source type: Reference]

[1]

Examples

.The Maclaurin series for any polynomial is the polynomial itself.^ The Taylor series for a polynomial is just the polynomial itself.
  • Taylor Series and Maclaurin Series, AP Calculus II/BC - Educator.com 20 January 2010 23:45 UTC www.educator.com [Source type: General]

^ The Maclaurin series for a polynomial is the polynomial: .

The Maclaurin series for (1 − x)−1 is the geometric series
1+x+x^2+x^3+\cdots\!
so the Taylor series for x−1 at a = 1 is
1-(x-1)+(x-1)^2-(x-1)^3+\cdots.\!
By integrating the above Maclaurin series we find the Maclaurin series for −ln(1 − x), where ln denotes the natural logarithm:
x+\frac{x^2}2+\frac{x^3}3+\frac{x^4}4+\cdots\!
and the corresponding Taylor series for ln(x) at a = 1 is
(x-1)-\frac{(x-1)^2}2+\frac{(x-1)^3}3-\frac{(x-1)^4}4+\cdots.\!
The Taylor series for the exponential function ex at a = 0 is
1 + \frac{x^1}{1!} + \frac{x^2}{2!} + \frac{x^3}{3!} + \frac{x^4}{4!} + \frac{x^5}{5!}+ \cdots \quad = \quad 1 + x + \frac{x^2}{2} + \frac{x^3}{6} + \frac{x^4}{24} + \frac{x^5}{120} + \cdots.\!
.The above expansion holds because the derivative of ex with respect to x is also ex and e0 equals 1. This leaves the terms (x − 0)n in the numerator and n!^ If there are still no nonzero terms, taylor doubles the degree of the expansion of g ( x ) so long as the degree of the expansion is less than or equal to n 2^taylordepth .

^ This is because, the higher order terms all have powers of x greater than 1, which means that the numerator is already much smaller than x , and then they are divided by larger and larger denominators.

^ The combination T < F > can be used to Taylor expand the solution of an ODE, while computing derivatives of the coefficients with respect to the point of expansion.
  • FADBAD++ 2.0 Automatic Differentiation Made Easy 19 September 2009 13:16 UTC www.fadbad.com [Source type: Reference]

in the denominator for each term in the infinite sum.

Convergence

The sine function (blue) is closely approximated by its Taylor polynomial of degree 7 (pink) for a full period centered at the origin.
.
The Taylor polynomials for log(1+x) only provide accurate approximations in the range −1 < x ≤ 1.
^ Taylor polynomials are usually good approximations more .
  • Taylor Polynomials and Series 20 January 2010 23:45 UTC www.slideshare.net [Source type: Reference]

^ Therefore, when a Taylor Polynomial can be substituted for a more complicated function in a wide range of mathematical problems.
  • Taylor Series - Math Images 20 January 2010 23:45 UTC mathforum.org [Source type: FILTERED WITH BAYES]

^ We're trying to approximate the function with a Taylor polynomial about 0 (i.e., a Maclaurin polynomial).
  • Taylor Series as Approximations - www.norsemathology.org 20 January 2010 23:45 UTC norsemathology.org [Source type: Reference]

Note that, for x > 1, the Taylor polynomials of higher degree are worse approximations.
.Taylor series need not in general be convergent.^ The general formula for a Taylor Series centered at x = a is: .
  • Taylor Series@Everything2.com 20 January 2010 23:45 UTC www.everything2.com [Source type: General]

^ To find the Taylor Series for a function we will need to determine a general formula for .
  • http://tutorial.math.lamar.edu/Classes/CalcII/TaylorSeries.aspx 20 January 2010 23:45 UTC tutorial.math.lamar.edu [Source type: FILTERED WITH BAYES]

^ General principles on accuracy of Taylor Series: .
  • Taylor Polynomial Applications, AP Calculus II/BC - Educator.com 20 January 2010 23:45 UTC www.educator.com [Source type: Reference]

.More precisely, the set of functions with a convergent Taylor series is a meager set in the Frechet space of smooth functions.^ Introduction to Taylor series as approximations to a function.
  • Math Tools Browse 20 January 2010 23:45 UTC mathforum.org [Source type: General]

^ CHAPTER 23 Chapter 23 - More on Taylor Series .

^ Suppose we want the Taylor series at 0 of the function .
  • Calculus/Taylor series - Wikibooks, collection of open-content textbooks 20 January 2010 23:45 UTC en.wikibooks.org [Source type: Reference]

.In spite of this, for many functions that arise in practice, the Taylor series does converge.^ Introduction to Taylor series as approximations to a function.
  • Math Tools Browse 20 January 2010 23:45 UTC mathforum.org [Source type: General]

^ Function: REVERT (expression,variable) Does reversion of Taylor Series.
  • Maxima Manual - Series 20 January 2010 23:45 UTC www.ma.utexas.edu [Source type: Reference]

^ Suppose we want the Taylor series at 0 of the function .
  • Calculus/Taylor series - Wikibooks, collection of open-content textbooks 20 January 2010 23:45 UTC en.wikibooks.org [Source type: Reference]

.The limit of a convergent Taylor series of a function f need not in general be equal to the function value f(x), but in practice often it is.^ The general formula for a Taylor Series centered at x = a is: .
  • Taylor Series@Everything2.com 20 January 2010 23:45 UTC www.everything2.com [Source type: General]

^ Introduction to Taylor series as approximations to a function.
  • Math Tools Browse 20 January 2010 23:45 UTC mathforum.org [Source type: General]

^ When a series converges absolutely, the corresponding series of absolute values converges.
  • Taylor Series Approximations 20 January 2010 23:45 UTC www.math.unh.edu [Source type: General]

For example, the function
 f(x) = \begin{cases} e^{-1/x^2}&\mathrm{if}\ x
ot=0\ 0&\mathrm{if}\ x=0 \end{cases}
is .infinitely differentiable at x = 0, and has all derivatives zero there.^ Since the limit the infinite term in this series goes to zero as n goes to infinity, then the series is convergent for all values of x.

^ There are then three cases for : either is infinite, is zero, or is some finite positive number greater than 0.
  • General Mathematical Identities for Analytic Functions: Series representations 20 January 2010 23:45 UTC functions.wolfram.com [Source type: Reference]

^ In contrast with the complex case, it turns out that all holomorphic functions are infinitely differentiable and have Taylor series that converge to them.
  • PlanetMath: Taylor series 19 September 2009 13:16 UTC planetmath.org [Source type: Reference]

.Consequently, the Taylor series of f(x) is zero.^ So The radius of convergence is 1 The interval of convergence is the open interval (−1, 1) Famous Taylor Series Example Compute Taylor series centered at zero for the following functions: ex sin x cos x (1 + x)p Example Compute the Taylor series centered at zero for f (x) = e x Example Compute the Taylor series centered at zero for f (x) = e x Solution f (x) = e x f (0) = 1 x f (x) = e f (0) = 1 x f (x) = e f (0) = 1 x f (x) = e f (0) = 1 ...
  • Taylor Polynomials and Series 20 January 2010 23:45 UTC www.slideshare.net [Source type: Reference]

^ This is a check, that tells us that the last terms get closer and closer to zero, and hence the Taylor series will be defined for all x.

^ GOLOMB, M. Zeros and poles of functions defined by Taylor series.
  • Solving Ordinary Differential Equations Using Taylor Series 20 January 2010 23:45 UTC doi.acm.org [Source type: Academic]

.However, f(x) is not equal to the zero function, and so it is not equal to its Taylor series.^ Taylor's series and approximation to analytic functions.

^ To conclude the Taylor Series for a function of x is: .

^ Suppose we want the Taylor series at 0 of the function .
  • Calculus/Taylor series - Wikibooks, collection of open-content textbooks 20 January 2010 23:45 UTC en.wikibooks.org [Source type: Reference]

.If f(x) is equal to its Taylor series in a neighborhood of a, it is said to be analytic in this neighborhood.^ Taylor's series and approximation to analytic functions.

^ If , this series converges only at the point , and the Taylor series offers little analytical benefit.
  • General Mathematical Identities for Analytic Functions: Series representations 20 January 2010 23:45 UTC functions.wolfram.com [Source type: Reference]

^ The sign ↔ means that the Taylor series should not be taken as purely equal to , since it may not converge everywhere on the complex plane.
  • General Mathematical Identities for Analytic Functions: Series representations 20 January 2010 23:45 UTC functions.wolfram.com [Source type: Reference]

.If f(x) is equal to its Taylor series everywhere it is called entire.^ Taylor has also booked a series regular of a pilot presentation called Which Way Is Up?
  • Taylor Lautner Fan [www.taylor-lautner.com] 20 January 2010 23:45 UTC taylor-lautner.com [Source type: General]

^ It is also not necessarily true that a Taylor series about $a$ equals the Taylor series of $f$ about some other point $b$ , when considered as functions.
  • PlanetMath: Taylor series 19 September 2009 13:16 UTC planetmath.org [Source type: Reference]

^ Okay, let's take just the first term in the Taylor Series, called the order approximation, and see what happens: So, the Taylor Series approximation says that if I must approximate with a constant, the best approximation is the constant = .
  • Taylor Series Approximations 20 January 2010 23:45 UTC www.math.unh.edu [Source type: General]

.The exponential function ex and the trigonometric functions sine and cosine are examples of entire functions.^ This solution is remarkable .It allows us to define the Sine and Cosine functions mathematically in terms of an infinite series.

^ Many problems in physics and engineering involve some function of an angle--most frequently a sine or a cosine function.
  • Taylor Series - Math Images 20 January 2010 23:45 UTC mathforum.org [Source type: FILTERED WITH BAYES]

^ Okay, we now need to work some examples that don’t involve the exponential function since these will tend to require a little more work.
  • http://tutorial.math.lamar.edu/Classes/CalcII/TaylorSeries.aspx 20 January 2010 23:45 UTC tutorial.math.lamar.edu [Source type: FILTERED WITH BAYES]

.Examples of functions that are not entire include the logarithm, the trigonometric function tangent, and its inverse arctan.^ In such cases it either represents an entire transcendental function—as, for example, the series does for the exponential function —or it contains only a finite number of terms and therefore represents a polynomial.
  • General Mathematical Identities for Analytic Functions: Series representations 20 January 2010 23:45 UTC functions.wolfram.com [Source type: Reference]

^ Choosing gives the Puiseux series for algebraic bivariate functions (because such functions should not include logarithms like ).
  • General Mathematical Identities for Analytic Functions: Series representations 20 January 2010 23:45 UTC functions.wolfram.com [Source type: Reference]

^ These include singular solutions to differential equations , a change of variables formula, and a way of relating the derivative of a function to the derivative of the inverse function.

.For these functions the Taylor series do not converge if x is far from a.^ Introduction to Taylor series as approximations to a function.
  • Math Tools Browse 20 January 2010 23:45 UTC mathforum.org [Source type: General]

^ Suppose we want the Taylor series at 0 of the function .
  • Calculus/Taylor series - Wikibooks, collection of open-content textbooks 20 January 2010 23:45 UTC en.wikibooks.org [Source type: Reference]

^ Is the Taylor series really that far off of the Math.exp() function?
  • Taylor Series ?? - CodeGuru Forums 20 January 2010 23:45 UTC www.codeguru.com [Source type: General]

.Taylor series can be used to calculate the value of an entire function in every point, if the value of the function, and of all of its derivatives, are known at a single point.^ For a function of a single variable, , the Taylor Series, , about a point is given by .
  • Taylor Series - Math Images 20 January 2010 23:45 UTC mathforum.org [Source type: FILTERED WITH BAYES]

^ Using this fact, we can write the Taylor Series of .
  • Taylor Series - Math Images 20 January 2010 23:45 UTC mathforum.org [Source type: FILTERED WITH BAYES]

^ Using the first seven derivatives we write the following Taylor series: .

Uses of the Taylor series for entire functions include:
.
  1. The partial sums (the Taylor polynomials) of the series can be used as approximations of the entire function.^ MathSciNet.   A summability approximation theorem for Taylor series of meromorphic functions.

    ^ Suppose we want the Taylor series at 0 of the function .
    • Calculus/Taylor series - Wikibooks, collection of open-content textbooks 20 January 2010 23:45 UTC en.wikibooks.org [Source type: Reference]

    ^ Taylor series and polynomials for functions of several variables .
    • PlanetMath: Taylor series 19 September 2009 13:16 UTC planetmath.org [Source type: Reference]

    These approximations are good if sufficiently many terms are included.
  2. The series representation simplifies many mathematical proofs.
Pictured on the right is an accurate approximation of sin(x) around the point a = 0. The pink curve is a polynomial of degree seven:
\sin\left( x \right) \approx x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!}.\!
.The error in this approximation is no more than |x|9/9!.^ To be consistent throughout this tutorial, we will define an interval where the truncated Taylor Series gives a good approximation for this function; so that the difference between the approximation and the original function is no more than throughout the interval.
  • Taylor Series Approximations 20 January 2010 23:45 UTC www.math.unh.edu [Source type: General]

^ To obtain a more uniform approximation over some interval in , other kinds of error criteria may be employed.
  • Taylor Series with Remainder 20 January 2010 23:45 UTC www.dsprelated.com [Source type: Reference]

^ As you can see, the parabola produced by using the order approximation of the Taylor Series approximates the function within a region around more accurately than the tangent line.
  • Taylor Series Approximations 20 January 2010 23:45 UTC www.math.unh.edu [Source type: General]

In particular, for −1 < x < 1, the error is less than 0.000003.
.In contrast, also shown is a picture of the natural logarithm function log(1 + x) and some of its Taylor polynomials around a = 0. These approximations converge to the function only in the region −1 < x ≤ 1; outside of this region the higher-degree Taylor polynomials are worse approximations for the function.^ The degree Taylor polynomial and the degree Taylor polynomial are equal: .

^ Find the degree Taylor polynomial for at .

^ Exponential function and natural logarithm : .
  • Calculus/Taylor series - Wikibooks, collection of open-content textbooks 20 January 2010 23:45 UTC en.wikibooks.org [Source type: Reference]

This is similar to Runge's phenomenon.
.The error incurred in approximating a function by its nth-degree Taylor polynomial, is called the remainder or residual and is denoted by the function Rn(x).^ The degree Taylor polynomial and the degree Taylor polynomial are equal: .

^ Find the degree Taylor polynomial for at .

^ Taylor series are great approximations of complicated functions using polynomials.
  • Taylor series for sine or sinus 20 January 2010 23:45 UTC dotancohen.com [Source type: General]

.Taylor's theorem can be used to obtain a bound on the size of the remainder.^ To check whether the series converges towards f ( x ), one normally uses estimates for the remainder term of Taylor's theorem .
  • Calculus/Taylor series - Wikibooks, collection of open-content textbooks 20 January 2010 23:45 UTC en.wikibooks.org [Source type: Reference]

^ The answer can be obtained from the Binomial expansion, but let us use Taylor series to show the relation between the binomial theorem and Taylor series.

^ READ ME! ( PDF version ) Remark on the logic of the proof ( PDF version ) This argument could be generalized to prove the simplest and most useful version of Taylor's theorem with remainder .
  • Taylor expansions 20 January 2010 23:45 UTC www.math.tamu.edu [Source type: FILTERED WITH BAYES]

History

.The Greek philosopher Zeno considered the problem of summing an infinite series to achieve a finite result, but rejected it as an impossibility: the result was Zeno's paradox.^ Then the dreaded Taylor Series formula tells us that The problem with this formula is that it's impossible to picture an infinte number of functions added together.
  • Taylor Series Approximations 20 January 2010 23:45 UTC www.math.unh.edu [Source type: General]

^ Infinite Taylor series expansion can be approximated by the truncated version, the finite Taylor polynomial expansions.
  • General Mathematical Identities for Analytic Functions: Series representations 20 January 2010 23:45 UTC functions.wolfram.com [Source type: Reference]

.Later, Aristotle proposed a philosophical resolution of the paradox, but the mathematical content was apparently unresolved until taken up by Democritus and then Archimedes.^ Well, that's just fine and dandy until later on down the line, in whatever career path you choose, it creeps back up on you.
  • Taylor Series Approximations 20 January 2010 23:45 UTC www.math.unh.edu [Source type: General]

.It was through Archimedes's method of exhaustion that an infinite number of progressive subdivisions could be performed to achieve a finite result.^ There are then three cases for : either is infinite, is zero, or is some finite positive number greater than 0.
  • General Mathematical Identities for Analytic Functions: Series representations 20 January 2010 23:45 UTC functions.wolfram.com [Source type: Reference]

[2] Liu Hui independently employed a similar method a few centuries later.[3]
.In the 14th century, the earliest examples of the use of Taylor series and closely-related methods were given by Madhava of Sangamagrama.^ Example 6 Find the Taylor Series for about .
  • http://tutorial.math.lamar.edu/Classes/CalcII/TaylorSeries.aspx 20 January 2010 23:45 UTC tutorial.math.lamar.edu [Source type: FILTERED WITH BAYES]

^ Using this fact, we can write the Taylor Series of .
  • Taylor Series - Math Images 20 January 2010 23:45 UTC mathforum.org [Source type: FILTERED WITH BAYES]

^ Example 9 Find the Taylor Series for about .
  • http://tutorial.math.lamar.edu/Classes/CalcII/TaylorSeries.aspx 20 January 2010 23:45 UTC tutorial.math.lamar.edu [Source type: FILTERED WITH BAYES]

[4] .Though no record of his work survives, writings of later Indian mathematicians suggest that he found a number of special cases of the Taylor series, including those for the trigonometric functions of sine, cosine, tangent, and arctangent.^ Taylor's series and approximation to analytic functions.

^ Now writing the Taylor Series: .

^ In very particular cases, q ‐series can coincide with some trigonometric functions.
  • General Mathematical Identities for Analytic Functions: Series representations 20 January 2010 23:45 UTC functions.wolfram.com [Source type: Reference]

.The Kerala school of astronomy and mathematics further expanded his works with various series expansions and rational approximations until the 16th century.^ MathSciNet.   Evaluation of higher-order semiclassical phase integrals: application of Chebyshev series approximation and Taylor series expansion.

^ MathSciNet.   A further example on the convergence of Taylor series    Glaister, P. Mathematics and computer education, 1996, vol.

^ Infinite Taylor series expansion can be approximated by the truncated version, the finite Taylor polynomial expansions.
  • General Mathematical Identities for Analytic Functions: Series representations 20 January 2010 23:45 UTC functions.wolfram.com [Source type: Reference]

.In the 17th century, James Gregory also worked in this area and published several Maclaurin series.^ Taylor series were actually discovered by James Gregory , who published Taylor series for functions and Maclaurin series for tan x, sec x, arctan x and arcsec x.
  • Taylor Series@Everything2.com 20 January 2010 23:45 UTC www.everything2.com [Source type: General]

^ Then in 1715, Brook Taylor came along and published Methodus incrementorum directa et inversa , repeating Gregory's earlier work.
  • Taylor Series@Everything2.com 20 January 2010 23:45 UTC www.everything2.com [Source type: General]

^ It includes personal records detailing her life and social commitments, correspondence with lesbian writers and friends as well as her publishers, and manuscripts of several of her published and unpublished works.

.It was not until 1715 however that a general method for constructing these series for all functions for which they exist was finally provided by Brook Taylor,[5] after whom the series are now named.^ Now writing the Taylor Series: .

^ So, provided a power series representation for the function about exists the Taylor Series for about is, .
  • http://tutorial.math.lamar.edu/Classes/CalcII/TaylorSeries.aspx 20 January 2010 23:45 UTC tutorial.math.lamar.edu [Source type: FILTERED WITH BAYES]

^ However, they do not exist in Version 2.0 now shipping on Windows.
  • Graphing Calculator 1.0 secrets 19 September 2009 13:16 UTC www.pacifict.com [Source type: Reference]

.The Maclaurin series was named after Colin Maclaurin, a professor in Edinburgh, who published the special case of the Taylor result in the 18th century.^ Case: GS Series Deluxe hardshell Wood & Steel Feature Bigger and bolder than a Taylor Grand Auditorium, it's the Grand Symphony series!
  • Taylor GS Series Grand Symphony from zZounds.com 20 January 2010 23:45 UTC www.zzounds.com [Source type: General]

^ The business records in the Series include contracts with publishers and a notebook in which Ms. Taylor recorded the submission of her literary manuscripts to publishers.

^ The special type of series known as Taylor series, allow us to express any mathematical function, real or complex, in terms of its n derivatives.

Properties

The function e−1/x² is not analytic at x = 0: the Taylor series is identically 0, although the function is not.
.If this series converges for every x in the interval (ar, a + r) and the sum is equal to f(x), then the function f(x) is said to be analytic in the interval (ar, a + r).^ So The radius of convergence is 1 The interval of convergence is the open interval (−1, 1) Famous Taylor Series Example Compute Taylor series centered at zero for the following functions: ex sin x cos x (1 + x)p Example Compute the Taylor series centered at zero for f (x) = e x Example Compute the Taylor series centered at zero for f (x) = e x Solution f (x) = e x f (0) = 1 x f (x) = e f (0) = 1 x f (x) = e f (0) = 1 x f (x) = e f (0) = 1 ...
  • Taylor Polynomials and Series 20 January 2010 23:45 UTC www.slideshare.net [Source type: Reference]

^ If our function was straight, then the first two terms of the Taylor Series would approximate the function sufficiently on a larger interval.
  • Taylor Series Approximations 20 January 2010 23:45 UTC www.math.unh.edu [Source type: General]

^ Taylor's series and approximation to analytic functions.

If this is true for any r then the function is said to be an entire function. .To check whether the series converges towards f(x), one normally uses estimates for the remainder term of Taylor's theorem.^ Here is the Taylor Series for this one.
  • http://tutorial.math.lamar.edu/Classes/CalcII/TaylorSeries.aspx 20 January 2010 23:45 UTC tutorial.math.lamar.edu [Source type: FILTERED WITH BAYES]

^ This is so when the Taylor series satisfies the "alternating series estimation theorem" (Stewart, p.
  • Taylor expansions 20 January 2010 23:45 UTC www.math.tamu.edu [Source type: FILTERED WITH BAYES]

^ Taylor series converges to the original function.
  • PlanetMath: Taylor series 19 September 2009 13:16 UTC planetmath.org [Source type: Reference]

.A function is analytic if and only if it can be represented as a power series; the coefficients in that power series are then necessarily the ones given in the above Taylor series formula.^ So, provided a power series representation for the function about exists the Taylor Series for about is, .
  • http://tutorial.math.lamar.edu/Classes/CalcII/TaylorSeries.aspx 20 January 2010 23:45 UTC tutorial.math.lamar.edu [Source type: FILTERED WITH BAYES]

^ The general formula for a Taylor Series centered at x = a is: .
  • Taylor Series@Everything2.com 20 January 2010 23:45 UTC www.everything2.com [Source type: General]

^ Introduction to Taylor series as approximations to a function.
  • Math Tools Browse 20 January 2010 23:45 UTC mathforum.org [Source type: General]

.The importance of such a power series representation is at least fourfold.^ So, provided a power series representation for the function about exists the Taylor Series for about is, .
  • http://tutorial.math.lamar.edu/Classes/CalcII/TaylorSeries.aspx 20 January 2010 23:45 UTC tutorial.math.lamar.edu [Source type: FILTERED WITH BAYES]

^ Now that we’ve assumed that a power series representation exists we need to determine what the coefficients, c n , are.
  • http://tutorial.math.lamar.edu/Classes/CalcII/TaylorSeries.aspx 20 January 2010 23:45 UTC tutorial.math.lamar.edu [Source type: FILTERED WITH BAYES]

^ Other power series have dense sets of singular points on the circle, such as the series , which has many singular points on the unit circle, the edge of its natural region of analyticity.
  • General Mathematical Identities for Analytic Functions: Series representations 20 January 2010 23:45 UTC functions.wolfram.com [Source type: Reference]

.First, differentiation and integration of power series can be performed term by term and is hence particularly easy.^ If our function was straight, then the first two terms of the Taylor Series would approximate the function sufficiently on a larger interval.
  • Taylor Series Approximations 20 January 2010 23:45 UTC www.math.unh.edu [Source type: General]

^ So far, all we have done is regenerate the first two terms in the Taylor Series.
  • Taylor Series Approximations 20 January 2010 23:45 UTC www.math.unh.edu [Source type: General]

^ So, lo and behold, the first two terms of the Taylor Series gives us the equation of the line tangent to our function at the point .
  • Taylor Series Approximations 20 January 2010 23:45 UTC www.math.unh.edu [Source type: General]

.Second, an analytic function can be uniquely extended to a holomorphic function defined on an open disk in the complex plane, which makes the whole machinery of complex analysis available.^ If some functions are not defined for the underlying arithmetic type then it is possible to make a dummy implementation that just throws an exception.
  • FADBAD++ 2.0 Automatic Differentiation Made Easy 19 September 2009 13:16 UTC www.fadbad.com [Source type: Reference]

^ If all coefficients , this Dirichlet series for the function converges in some open half-plane .
  • General Mathematical Identities for Analytic Functions: Series representations 20 January 2010 23:45 UTC functions.wolfram.com [Source type: Reference]

^ If , the sum of the series defines a regular analytic function having at least one singular point on the circle .
  • General Mathematical Identities for Analytic Functions: Series representations 20 January 2010 23:45 UTC functions.wolfram.com [Source type: Reference]

.Third, the (truncated) series can be used to compute function values approximately (often by recasting the polynomial into the Chebyshev form and evaluating it with the Clenshaw algorithm).^ Introduction to Taylor series as approximations to a function.
  • Math Tools Browse 20 January 2010 23:45 UTC mathforum.org [Source type: General]

^ Introduces Taylor polynomials as approximation to functions.
  • Math Tools Browse 20 January 2010 23:45 UTC mathforum.org [Source type: General]

^ Taylor's series and approximation to analytic functions.

.Fourth, algebraic operations can often be done much more readily on the power series representation; for instance the simplest proof of Euler's formula uses the Taylor series expansions for sine, cosine, and exponential functions.^ Let's look at the Taylor expansion of this function: .

^ The general formula for a Taylor Series centered at x = a is: .
  • Taylor Series@Everything2.com 20 January 2010 23:45 UTC www.everything2.com [Source type: General]

^ This exponential Fourier series expansion can be transformed into a trigonometric Fourier series by using the Euler formula .
  • General Mathematical Identities for Analytic Functions: Series representations 20 January 2010 23:45 UTC functions.wolfram.com [Source type: Reference]

This result is of fundamental importance in such fields as harmonic analysis.
.Another reason why the Taylor series is the natural power series for studying a function f is that, given the value of f and its derivatives at a point a, the Taylor series is in some sense the most likely function that fits the given data.^ So, provided a power series representation for the function about exists the Taylor Series for about is, .
  • http://tutorial.math.lamar.edu/Classes/CalcII/TaylorSeries.aspx 20 January 2010 23:45 UTC tutorial.math.lamar.edu [Source type: FILTERED WITH BAYES]

^ However, if we take the derivative of the function (and its power series) then plug in we get, .
  • http://tutorial.math.lamar.edu/Classes/CalcII/TaylorSeries.aspx 20 January 2010 23:45 UTC tutorial.math.lamar.edu [Source type: FILTERED WITH BAYES]

^ If this series converges, it coincides with the Taylor power series expansion at the point .
  • General Mathematical Identities for Analytic Functions: Series representations 20 January 2010 23:45 UTC functions.wolfram.com [Source type: Reference]

[6]
.Note that there are examples of infinitely differentiable functions f(x) whose Taylor series converge, but are not equal to f(x).^ Example 6 Find the Taylor Series for about .
  • http://tutorial.math.lamar.edu/Classes/CalcII/TaylorSeries.aspx 20 January 2010 23:45 UTC tutorial.math.lamar.edu [Source type: FILTERED WITH BAYES]

^ Example 9 Find the Taylor Series for about .
  • http://tutorial.math.lamar.edu/Classes/CalcII/TaylorSeries.aspx 20 January 2010 23:45 UTC tutorial.math.lamar.edu [Source type: FILTERED WITH BAYES]

^ Example 5 Find the Taylor Series for about .
  • http://tutorial.math.lamar.edu/Classes/CalcII/TaylorSeries.aspx 20 January 2010 23:45 UTC tutorial.math.lamar.edu [Source type: FILTERED WITH BAYES]

.For instance, the function defined pointwise by f(x) = e−1/x² if x ≠ 0 and f(0) = 0 is an example of a non-analytic smooth function.^ If , the sum of the series defines a regular analytic function having at least one singular point on the circle .
  • General Mathematical Identities for Analytic Functions: Series representations 20 January 2010 23:45 UTC functions.wolfram.com [Source type: Reference]

^ In more complicated cases, q ‐series can define different special functions, for example elliptic theta functions: .
  • General Mathematical Identities for Analytic Functions: Series representations 20 January 2010 23:45 UTC functions.wolfram.com [Source type: Reference]

^ If , then the Laurent series converges absolutely in the ring , where its sum defines an analytic function.
  • General Mathematical Identities for Analytic Functions: Series representations 20 January 2010 23:45 UTC functions.wolfram.com [Source type: Reference]

.All its derivatives at x = 0 are zero, so the Taylor series of f(x) at 0 is zero everywhere, even though the function is nonzero for every x ≠ 0. This particular pathology does not afflict Taylor series in complex analysis.^ GOLOMB, M. Zeros and poles of functions defined by Taylor series.
  • Solving Ordinary Differential Equations Using Taylor Series 20 January 2010 23:45 UTC doi.acm.org [Source type: Academic]

^ Introduction to Taylor series as approximations to a function.
  • Math Tools Browse 20 January 2010 23:45 UTC mathforum.org [Source type: General]

^ Fact The Taylor series for the function f (x) = e x converges for all x to ex .
  • Taylor Polynomials and Series 20 January 2010 23:45 UTC www.slideshare.net [Source type: Reference]

.There, the area of convergence of a Taylor series is always a disk in the complex plane (possibly with radius 0), and where the Taylor series converges, it converges to the function value.^ Introduction to Taylor series as approximations to a function.
  • Math Tools Browse 20 January 2010 23:45 UTC mathforum.org [Source type: General]

^ The series converges absolutely in some disk of radius centered on , where is called the radius of convergence .
  • General Mathematical Identities for Analytic Functions: Series representations 20 January 2010 23:45 UTC functions.wolfram.com [Source type: Reference]

^ When a series converges absolutely, the corresponding series of absolute values converges.
  • Taylor Series Approximations 20 January 2010 23:45 UTC www.math.unh.edu [Source type: General]

Notice that e−1/z² does not approach 0 as z approaches 0 along the imaginary axis, hence this function is not continuous as a function on the complex plane.
.Since every sequence of real or complex numbers can appear as coefficients in the Taylor series of an infinitely differentiable function defined on the real line,[Proof] the radius of convergence of a Taylor series can be zero.^ Thus, the radius of convergence of our function is infinite.
  • Taylor Series Approximations 20 January 2010 23:45 UTC www.math.unh.edu [Source type: General]

^ GOLOMB, M. Zeros and poles of functions defined by Taylor series.
  • Solving Ordinary Differential Equations Using Taylor Series 20 January 2010 23:45 UTC doi.acm.org [Source type: Academic]

^ R is called the radius of convergence of the power series.
  • Taylor Polynomials and Series 20 January 2010 23:45 UTC www.slideshare.net [Source type: Reference]

.There are even infinitely differentiable functions defined on the real line whose Taylor series have a radius of convergence 0 everywhere.^ Thus, the radius of convergence of our function is infinite.
  • Taylor Series Approximations 20 January 2010 23:45 UTC www.math.unh.edu [Source type: General]

^ GOLOMB, M. Zeros and poles of functions defined by Taylor series.
  • Solving Ordinary Differential Equations Using Taylor Series 20 January 2010 23:45 UTC doi.acm.org [Source type: Academic]

^ R is called the radius of convergence of the power series.
  • Taylor Polynomials and Series 20 January 2010 23:45 UTC www.slideshare.net [Source type: Reference]

[7]
.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 also negative powers of the variable x; see Laurent series.^ The coefficient of the power in the Laurent expansion of function is called the residue of at the point : .
  • General Mathematical Identities for Analytic Functions: Series representations 20 January 2010 23:45 UTC functions.wolfram.com [Source type: Reference]

^ In very particular cases, q ‐series can coincide with some trigonometric functions.
  • General Mathematical Identities for Analytic Functions: Series representations 20 January 2010 23:45 UTC functions.wolfram.com [Source type: Reference]

^ The power series , for example, has only one singular point, at .
  • General Mathematical Identities for Analytic Functions: Series representations 20 January 2010 23:45 UTC functions.wolfram.com [Source type: Reference]

For example, f(x) = e−1/x² can be written as a Laurent series.

List of Maclaurin series of some common functions

See also List of mathematical series
The real part of the cosine function in the complex plane.
An 8th degree approximation of the cosine function in the complex plane.
The two above curves put together.
.Several important Maclaurin series expansions follow.^ Some Interesting Series Resulting from a Certain MacLaurin Expansion    M.R.Spiegel   The American Mathematical Monthly,Vol.60,No.4.

[8] All these expansions are valid for complex arguments x.
\mathrm{e}^{x} = \sum^{\infin}_{n=0} \frac{x^n}{n!} = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \cdots	ext{ for all } x\!
\ln(1-x) = -\sum^{\infin}_{n=1} \frac{x^n}n	ext{ for } -1\le x<1
\ln(1+x) = \sum^{\infin}_{n=1} (-1)^{n+1}\frac{x^n}n	ext{ for }-1<x\le1
\frac{1-x^{m + 1}}{1-x} = \sum^{m}_{n=0} x^n\quad\mbox{ for } x 
ot= 1	ext{ and } m\in\mathbb{N}_0\!
Infinite geometric series:
\frac{1}{1-x} = \sum^{\infin}_{n=0} x^n	ext{ for }|x| < 1\!
Variants of the infinite geometric series:
\frac{x^m}{1-x} = \sum^{\infin}_{n=m} x^n\quad\mbox{ for }|x| < 1 	ext{ and } m\in\mathbb{N}_0\!
\frac{x}{(1-x)^2} = \sum^{\infin}_{n=1}n x^n\quad	ext{ for }|x| < 1\!
\sqrt{1+x} = \sum_{n=0}^\infty \frac{(-1)^n(2n)!}{(1-2n)(n!)^2(4^n)}x^n = 1 + 	extstyle \frac{1}{2}x - \frac{1}{8}x^2 + \frac{1}{16} x^3 - \frac{5}{128} x^4 + \dots	ext{ for }|x|<1\!
Binomial series (includes the square root for α = 1/2 and the infinite geometric series for α = −1):
(1+x)^\alpha = \sum_{n=0}^\infty {\alpha \choose n} x^n\quad\mbox{ for all }|x| < 1 	ext{ and all complex } \alpha\!
with generalized binomial coefficients
{\alpha\choose n} = \prod_{k=1}^n \frac{\alpha-k+1}k = \frac{\alpha(\alpha-1)\cdots(\alpha-n+1)}{n!}
\sin x = \sum^{\infin}_{n=0} \frac{(-1)^n}{(2n+1)!} x^{2n+1} = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \cdots	ext{ for all } x\!
\cos x = \sum^{\infin}_{n=0} \frac{(-1)^n}{(2n)!} x^{2n} = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \cdots	ext{ for all } x\!
	an x = \sum^{\infin}_{n=1} \frac{B_{2n} (-4)^n (1-4^n)}{(2n)!} x^{2n-1} = x + \frac{x^3}{3} + \frac{2 x^5}{15} + \cdots	ext{ for }|x| < \frac{\pi}{2}\!
where the Bs are Bernoulli numbers.
\sec x = \sum^{\infin}_{n=0} \frac{(-1)^n E_{2n}}{(2n)!} x^{2n}	ext{ for }|x| < \frac{\pi}{2}\!
\arcsin x = \sum^{\infin}_{n=0} \frac{(2n)!}{4^n (n!)^2 (2n+1)} x^{2n+1}	ext{ for }|x| \le 1\!
\arctan x = \sum^{\infin}_{n=0} \frac{(-1)^n}{2n+1} x^{2n+1}	ext{ for }|x| \le 1\!
\sinh x = \sum^{\infin}_{n=0} \frac{x^{2n+1}}{(2n+1)!} = x + \frac{x^3}{3!} + \frac{x^5}{5!} + \cdots	ext{ for all } x\!
\cosh x = \sum^{\infin}_{n=0} \frac{x^{2n}}{(2n)!} = 1 + \frac{x^2}{2!} + \frac{x^4}{4!} + \cdots	ext{ for all } x\!
	anh x = \sum^{\infin}_{n=1} \frac{B_{2n} 4^n (4^n-1)}{(2n)!} x^{2n-1} = x-\frac{1}{3}x^3+\frac{2}{15}x^5-\frac{17}{315}x^7+\cdots 	ext{ for }|x| < \frac{\pi}{2}\!
\mathrm{arsinh} (x) = \sum^{\infin}_{n=0} \frac{(-1)^n (2n)!}{4^n (n!)^2 (2n+1)} x^{2n+1}	ext{ for }|x| \le 1\!
\mathrm{artanh} (x) = \sum^{\infin}_{n=0} \frac{x^{2n+1}}{2n+1} 	ext{ for }|x| < 1\!
W_0(x) = \sum^{\infin}_{n=1} \frac{(-n)^{n-1}}{n!} x^n	ext{ for }|x| < \frac{1}{\mathrm{e}}\!
The numbers Bk appearing in the summation expansions of tan(x) and tanh(x) are the Bernoulli numbers. The Ek in the expansion of sec(x) are Euler numbers.

Calculation of Taylor series

.Several methods exist for the calculation of Taylor series of a large number of functions.^ Taylor's series and approximation to analytic functions.

^ Maxima contains functions taylor and powerseries for finding the series of differentiable functions.

^ MathSciNet.   A summability approximation theorem for Taylor series of meromorphic functions.

.One can attempt to use the Taylor series as-is and generalize the form of the coefficients, or one can use manipulations such as substitution, multiplication or division, addition or subtraction of standard Taylor series to construct the Taylor series of a function, by virtue of Taylor series being power series.^ Operations such as addition and multiplication work as usual on series.

^ This DAG can then be used to find the Taylor coefficients.
  • FADBAD++ 2.0 Automatic Differentiation Made Easy 19 September 2009 13:16 UTC www.fadbad.com [Source type: Reference]

^ Function: taylor_simplifier ( expr ) Simplifies coefficients of the power series expr .

.In some cases, one can also derive the Taylor series by repeatedly applying integration by parts.^ Case: GS Series Deluxe hardshell Wood & Steel Feature Bigger and bolder than a Taylor Grand Auditorium, it's the Grand Symphony series!
  • Taylor GS Series Grand Symphony from zZounds.com 20 January 2010 23:45 UTC www.zzounds.com [Source type: General]

^ MathSciNet.   Mellin transform methods applied to integral evaluation: Taylor series and asymptotic approximations.

^ People rated this product : 10 out of 10 4 People wrote reviews Read all Taylor GS Series Grand Symphony Acoustic Guitar (with Case) reviews...
  • Taylor GS Series Grand Symphony from zZounds.com 20 January 2010 23:45 UTC www.zzounds.com [Source type: General]

.Particularly convenient is the use of computer algebra systems to calculate Taylor series.^ Uses a power series to calculate the integral of arctan(x^2).
  • Math Tools Browse 20 January 2010 23:45 UTC mathforum.org [Source type: General]

^ Of course, the goal of this tutorial is to convince you that the Taylor Series is a useful tool to approximate your original function near your point.
  • Taylor Series Approximations 20 January 2010 23:45 UTC www.math.unh.edu [Source type: General]

^ So, let's use two functions in this demonstration to illustrate the Taylor Series approximation: a.
  • Taylor Series Approximations 20 January 2010 23:45 UTC www.math.unh.edu [Source type: General]

First example

Compute the 7th degree Maclaurin polynomial for the function
f(x)=\ln\cos x, \quad x\in(-\pi/2, \pi/2)\!.
First, rewrite the function as
f(x)=\ln(1+(\cos x-1))\!.
We have for the natural logarithm (by using the big O notation)
\ln(1+x) = x - \frac{x^2}2 + \frac{x^3}3 + {O}(x^4)\!
and for the cosine function
\cos x - 1 = -\frac{x^2}2 + \frac{x^4}{24} - \frac{x^6}{720} + {O}(x^8)\!
The latter series expansion has a zero constant term, which enables us to substitute the second series into the first one and to easily omit terms of higher order than the 7th degree by using the big O notation:
\begin{align}f(x)&=\ln(1+(\cos x-1))\ &=\bigl(\cos x-1\bigr) - \frac12\bigl(\cos x-1\bigr)^2 + \frac13\bigl(\cos x-1\bigr)^3+ {O}\bigl((\cos x-1)^4\bigr)\\&=\biggl(-\frac{x^2}2 + \frac{x^4}{24} - \frac{x^6}{720} +{O}(x^8)\biggr)-\frac12\biggl(-\frac{x^2}2+\frac{x^4}{24}+{O}(x^6)\biggr)^2+\frac13\biggl(-\frac{x^2}2+O(x^4)\biggr)^3 + {O}(x^8)\\ & =-\frac{x^2}2 + \frac{x^4}{24}-\frac{x^6}{720} - \frac{x^4}8 + \frac{x^6}{48} - \frac{x^6}{24} +O(x^8)\ & =- \frac{x^2}2 - \frac{x^4}{12} - \frac{x^6}{45}+O(x^8). \end{align}\!
.Since the cosine is an even function, the coefficients for all the odd powers x, x3, x5, x7, ...^ The DAG will instead hold all Taylor coefficients of degree 0...k of all intermediate variables for calculating the function.
  • FADBAD++ 2.0 Automatic Differentiation Made Easy 19 September 2009 13:16 UTC www.fadbad.com [Source type: Reference]

^ Even and Odd Functions .
  • Informal Derivation of Taylor Series 20 January 2010 23:45 UTC www.dsprelated.com [Source type: Reference]

have to be zero.

Second example

Suppose we want the Taylor series at 0 of the function
g(x)=\frac{e^x}{\cos x}\!.
We have for the exponential function
e^x = \sum^\infty_{n=0} {x^n\over n!} =1 + x + {x^2 \over 2!} + {x^3 \over 3!} + {x^4 \over 4!}+\cdots\!
and, as in the first example,
\cos x = 1 - {x^2 \over 2!} + {x^4 \over 4!} - \cdots\!
Assume the power series is
{e^x \over \cos x} = c_0 + c_1 x + c_2 x^2 + c_3 x^3 + \cdots\!
Then multiplication with the denominator and substitution of the series of the cosine yields
\begin{align} e^x &= (c_0 + c_1 x + c_2 x^2 + c_3 x^3 + \cdots)\cos x\ &=\left(c_0 + c_1 x + c_2 x^2 + c_3 x^3 + c_4x^4 + \cdots\right)\left(1 - {x^2 \over 2!} + {x^4 \over 4!} - \cdots\right)\\&=c_0 - {c_0 \over 2}x^2 + {c_0 \over 4!}x^4 + c_1x - {c_1 \over 2}x^3 + {c_1 \over 4!}x^5 + c_2x^2 - {c_2 \over 2}x^4 + {c_2 \over 4!}x^6 + c_3x^3 - {c_3 \over 2}x^5 + {c_3 \over 4!}x^7 +\cdots \end{align}\!
Collecting the terms up to fourth order yields
=c_0 + c_1x + \left(c_2 - {c_0 \over 2}\right)x^2 + \left(c_3 - {c_1 \over 2}\right)x^3+\left(c_4+{c_0 \over 4!}-{c_2\over 2}\right)x^4 + \cdots\!
Comparing coefficients with the above series of the exponential function yields the desired Taylor series
\frac{e^x}{\cos x}=1 + x + x^2 + {2x^3 \over 3} + {x^4 \over 2} + \cdots.\!

Taylor series as definitions

.Classically, algebraic functions are defined by an algebraic equation, and transcendental functions (including those discussed above) are defined by some property that holds for them, such as a differential equation.^ Jstor.   Taylor polynomials of implicit functions, of inverse functions, and of solutions of ordinary differential equations.

^ MathSciNet.   On the representation by integrals of some functions defined by Taylor expansions and its application to the solution of partial differential equations.

^ Taylor-Dirichlet Series and Algebraic Differential-Difference Equations    Frank Wadleigh   Proceedings of the American Mathematical Society, Vol.

For example the exponential function is the function which is equal to its own derivative everywhere, and assumes the value 1 at the origin. .However, one may equally well define an analytic function by its Taylor series.^ Taylor's series and approximation to analytic functions.

^ Introduction to Taylor series as approximations to a function.
  • Math Tools Browse 20 January 2010 23:45 UTC mathforum.org [Source type: General]

^ The function will continue performing the Taylor series calculation until the relative error is less than 10^-16 (generally).
  • Taylor Series ?? - CodeGuru Forums 20 January 2010 23:45 UTC www.codeguru.com [Source type: General]

.Taylor series are used to define functions and "operators" in diverse areas of mathematics.^ Taylor's series and approximation to analytic functions.

^ MathSciNet.   Approximate solution of linear systems with point delays using Taylor series.

^ Solving ordinary differential equations using Taylor series.

In particular, this is true in areas where the classical definitions of functions break down. .For example, using Taylor series, one may define analytical functions of matrices and operators, such as the matrix exponential or matrix logarithm.^ Introduction to Taylor series as approximations to a function.
  • Math Tools Browse 20 January 2010 23:45 UTC mathforum.org [Source type: General]

^ The function will continue performing the Taylor series calculation until the relative error is less than 10^-16 (generally).
  • Taylor Series ?? - CodeGuru Forums 20 January 2010 23:45 UTC www.codeguru.com [Source type: General]

^ Proof that Taylor polynomials are the appropriate series representation of a function.
  • Math Tools Browse 20 January 2010 23:45 UTC mathforum.org [Source type: General]

In other areas, such as formal analysis, it is more convenient to work directly with the power series themselves. .Thus one may define a solution of a differential equation as a power series which, one hopes to prove, is the Taylor series of the desired solution.^ Jstor.   Taylor polynomial solutions of Volterra integral equations.
  • Maclaurin and Taylor Series 20 January 2010 23:45 UTC math.fullerton.edu [Source type: Academic]

^ A Taylor series method for the numerical solution of two-point boundary value problems.
  • Maclaurin and Taylor Series 20 January 2010 23:45 UTC math.fullerton.edu [Source type: Academic]

^ Taylor series solution of a class of diffusion problem in physiology.
  • Maclaurin and Taylor Series 20 January 2010 23:45 UTC math.fullerton.edu [Source type: Academic]

Taylor series in several variables

The Taylor series may also be generalized to functions of more than one variable with
T(x_1,\cdots,x_d) =
=\sum_{n_1=0}^{\infin} \cdots \sum_{n_d=0}^{\infin} \frac{(x_1-a_1)^{n_1}\cdots (x_d-a_d)^{n_d}}{n_1!\cdots n_d!}\,\left(\frac{\partial^{n_1 + \cdots + n_d}f}{\partial x_1^{n_1}\cdots \partial x_d^{n_d}}\right)(a_1,\dots,a_d).\!
For example, for a function that depends on two variables, x and y, the Taylor series to second order about the point (a, b) is:
f(x,y)\!
\approx f(a,b) +(x-a)\, f_x(a,b) +(y-b)\, f_y(a,b) \!
+ \frac{1}{2!}\left[ (x-a)^2\,f_{xx}(a,b) + 2(x-a)(y-b)\,f_{xy}(a,b) +(y-b)^2\, f_{yy}(a,b) \right]\,,
where the subscripts denote the respective partial derivatives.
A second-order Taylor series expansion of a scalar-valued function of more than one variable can be written compactly as
T(\mathbf{x}) = f(\mathbf{a}) + (\mathbf{x} - \mathbf{a})^T\mathrm{D} f(\mathbf{a}) + \frac{1}{2!} (\mathbf{x} - \mathbf{a})^T \,\{\mathrm{D}^2 f(\mathbf{a})\}\,(\mathbf{x} - \mathbf{a}) + \cdots\! \,,
where D f(\mathbf{a})\! is the gradient of \,f evaluated at \mathbf{x} = \mathbf{a} and D^2 f(\mathbf{a})\! is the Hessian matrix. Applying the multi-index notation the Taylor series for several variables becomes
T(\mathbf{x}) = \sum_{|\alpha| \ge 0}^{}\frac{(\mathbf{x}-\mathbf{a})^{\alpha}}{\alpha !}\,({\mathrm{\partial}^{\alpha}}\,f)(\mathbf{a})\,,
which is to be understood as a still more abbreviated multi-index version of the first equation of this paragraph, again in full analogy to the single variable case.

See also

Notes

  1. ^ An Introduction to Mechanics, Kleppner and Kolenkow; ISBN 0-07-463685-5
  2. ^ Kline, M. (1990) Mathematical Thought from Ancient to Modern Times. Oxford University Press. pp. 35-37.
  3. ^ Boyer, C. and Merzbach, U. (1991) A History of Mathematics. John Wiley and Sons. pp. 202-203.
  4. ^ "Neither Newton nor Leibniz - The Pre-History of Calculus and Celestial Mechanics in Medieval Kerala". MAT 314. Canisius College. http://www.canisius.edu/topos/rajeev.asp. Retrieved 2006-07-09. 
  5. ^ Taylor, Brook, Methodus Incrementorum Directa et Inversa [Direct and Reverse Methods of Incrementation] (London, 1715), pages 21-23 (Proposition VII, Theorem 3, Corollary 2). Translated into English in D. J. Struik, A Source Book in Mathematics 1200-1800 (Cambridge, Massachusetts: Harvard University Press, 1969), pages 329-332.
  6. ^ Bruss, F. Thomas (1982), "A probabilistic approach to an approximation problem", Annales de la Societé Scientifique de Bruxelles. Série I. Sciences Mathématiques, Astronomiques et Physiques 96 (2): 91–97 .
  7. ^ Rudin, Walter (1980), Real and Complex Analysis, New Dehli: McGraw-Hill, p. 418, Exercise 13, ISBN 0-07-099557-5 
  8. ^ Most of these can be found in (Abramowitz & Stegun 1970).

References

.
  • Abramowitz, Milton; Stegun, Irene A. (1970), Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, New York: Dover Publications, Ninth printing 
  • Thomas, George B. Jr.; Finney, Ross L. (1996), Calculus and Analytic Geometry (9th ed.^ RATTAPALLAX #5, edited by George Dickerson (New York, 2001) Taylor poetry: TURNING 57 • LATE SHOW AT THE STARLIGHT LAUNDRY .

    ^ Net.   Maclaurin and Taylor Series for Transcendental Functions: A Graphing-Calculator View of Convergence    Stick, Marvin E. The mathematics teacher, 1999, vol.

    ^ RATTAPALLAX #4, edited by George Dickerson (New York) Taylor poetry: DRAWING BLOOD .

    )
    , Addison Wesley, ISBN 0-201-53174-7 
  • Greenberg, Michael (1998), Advanced Engineering Mathematics (2nd ed.), Prentice Hall, ISBN 0-13-321431-1 

External links


Simple English

In mathematics, a Taylor series shows a function as an infinite sum. The sum's terms are taken from the function's derivatives. Taylor series come from Taylor's theorem.

Definition

The Taylor Series of a function f(x) is defined by T(x):=\sum_{n=1}^{\infty}\frac{f^{(n)}(a)}{n!}(x-a)^{n}, where a is a number in the function's domain. If the Taylor Series of a function is equal to that function, we call the function analytic.


Citable sentences

Up to date as of December 17, 2010

Here are sentences from other pages on Taylor series, which are similar to those in the above article.








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