44-2042: [REDACTED] Look up Taylor in Wiktionary, the free dictionary. Taylor , Taylors or Taylor's may refer to: People [ edit ] Taylor (surname) List of people with surname Taylor Taylor (given name) , including Tayla and Taylah Taylor sept , a branch of Scottish clan Cameron Justice Taylor (disambiguation) Places [ edit ] Australia [ edit ] Electoral district of Taylor , South Australia Taylor, Australian Capital Territory , planned suburb Canada [ edit ] Taylor, British Columbia United States [ edit ] Taylor, Alabama Taylor, Arizona Taylor, Arkansas Taylor, Indiana Taylor, Louisiana Taylor, Maryland Taylor, Michigan Taylor, Mississippi Taylor, Missouri Taylor, Nebraska Taylor, North Dakota Taylor, New York Taylor, Beckham County, Oklahoma Taylor, Cotton County, Oklahoma Taylor, Pennsylvania Taylors, South Carolina Taylor, Texas Taylor, Utah Taylor, Washington Taylor, West Virginia Taylor, Wisconsin Taylor, Wyoming Taylor County (disambiguation) Taylor Township (disambiguation) Businesses and organisations [ edit ] Taylor's (department store) in Quebec, Canada Taylor Guitars , an American guitar manufacturer Taylor University , in Upland, Indiana, U.S. Taylor's University , commonly referred to as Taylor's, in Subang Jaya, Selangor, Malaysia Taylor's College John Taylor & Co , or Taylor's Bell Foundry, Taylor's of Loughborough, or Taylor's, in England Taylor Company ,
88-422: A ) n . {\displaystyle f(a)+{\frac {f'(a)}{1!}}(x-a)+{\frac {f''(a)}{2!}}(x-a)^{2}+{\frac {f'''(a)}{3!}}(x-a)^{3}+\cdots =\sum _{n=0}^{\infty }{\frac {f^{(n)}(a)}{n!}}(x-a)^{n}.} Here, n ! denotes the factorial of n . The function f ( a ) denotes the n th derivative of f evaluated at the point a . The derivative of order zero of f is defined to be f itself and ( x −
132-541: A i = e − u ∑ j = 0 ∞ u j j ! a i + j . {\displaystyle \sum _{n=0}^{\infty }{\frac {u^{n}}{n!}}\Delta ^{n}a_{i}=e^{-u}\sum _{j=0}^{\infty }{\frac {u^{j}}{j!}}a_{i+j}.} So in particular, f ( a + t ) = lim h → 0 + e − t / h ∑ j = 0 ∞ f (
176-434: A n ( x − b ) n . {\displaystyle f(x)=\sum _{n=0}^{\infty }a_{n}(x-b)^{n}.} Differentiating by x the above formula n times, then setting x = b gives: f ( n ) ( b ) n ! = a n {\displaystyle {\frac {f^{(n)}(b)}{n!}}=a_{n}} and so the power series expansion agrees with
220-457: A ) h n = f ( a + t ) . {\displaystyle \lim _{h\to 0^{+}}\sum _{n=0}^{\infty }{\frac {t^{n}}{n!}}{\frac {\Delta _{h}^{n}f(a)}{h^{n}}}=f(a+t).} Here Δ h is the n th finite difference operator with step size h . The series is precisely the Taylor series, except that divided differences appear in place of differentiation: the series
264-399: A ) + f ″ ( a ) 2 ! ( x − a ) 2 + f ‴ ( a ) 3 ! ( x − a ) 3 + ⋯ = ∑ n = 0 ∞ f ( n ) ( a ) n ! ( x −
308-454: A = 1 is ( x − 1 ) − 1 2 ( x − 1 ) 2 + 1 3 ( x − 1 ) 3 − 1 4 ( x − 1 ) 4 + ⋯ , {\displaystyle (x-1)-{\tfrac {1}{2}}(x-1)^{2}+{\tfrac {1}{3}}(x-1)^{3}-{\tfrac {1}{4}}(x-1)^{4}+\cdots ,} and more generally,
352-884: A ) and 0! are both defined to be 1 . This series can be written by using sigma notation , as in the right side formula. With a = 0 , the Maclaurin series takes the form: f ( 0 ) + f ′ ( 0 ) 1 ! x + f ″ ( 0 ) 2 ! x 2 + f ‴ ( 0 ) 3 ! x 3 + ⋯ = ∑ n = 0 ∞ f ( n ) ( 0 ) n ! x n . {\displaystyle f(0)+{\frac {f'(0)}{1!}}x+{\frac {f''(0)}{2!}}x^{2}+{\frac {f'''(0)}{3!}}x^{3}+\cdots =\sum _{n=0}^{\infty }{\frac {f^{(n)}(0)}{n!}}x^{n}.} The Taylor series of any polynomial
396-494: A few centuries later. In the 14th century, the earliest examples of specific Taylor series (but not the general method) were given by Indian mathematician Madhava of Sangamagrama . Though no record of his work survives, writings of his followers in the Kerala school of astronomy and mathematics suggest that he found the Taylor series for the trigonometric functions of sine , cosine , and arctangent (see Madhava series ). During
440-1005: A general method for expanding functions in series. Newton had in fact used a cumbersome method involving long division of series and term-by-term integration, but Gregory did not know it and set out to discover a general method for himself. In early 1671 Gregory discovered something like the general Maclaurin series and sent a letter to Collins including series for arctan x , {\textstyle \arctan x,} tan x , {\textstyle \tan x,} sec x , {\textstyle \sec x,} ln sec x {\textstyle \ln \,\sec x} (the integral of tan {\displaystyle \tan } ), ln tan 1 2 ( 1 2 π + x ) {\textstyle \ln \,\tan {\tfrac {1}{2}}{{\bigl (}{\tfrac {1}{2}}\pi +x{\bigr )}}} (the integral of sec ,
484-692: A maker of foodservice equipment owned by Middleby Corporation Science and technology [ edit ] Taylor's theorem , in calculus Taylor series , in mathematics AMD Taylor, alternate name for the Turion 64 X2 computer processor Taylor knock-out factor , for evaluating the stopping power of hunting cartridges Other uses [ edit ] Taylor Series , infinite series of polynomials which asymptotically approaches infinitely differentiable functions Taylor's law , an empirical law in ecology Taylor rule , in economics Taylor Law , an article of New York State Law Taylor (crater) , on
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#1732837989489528-644: A maker of foodservice equipment owned by Middleby Corporation Science and technology [ edit ] Taylor's theorem , in calculus Taylor series , in mathematics AMD Taylor, alternate name for the Turion 64 X2 computer processor Taylor knock-out factor , for evaluating the stopping power of hunting cartridges Other uses [ edit ] Taylor Series , infinite series of polynomials which asymptotically approaches infinitely differentiable functions Taylor's law , an empirical law in ecology Taylor rule , in economics Taylor Law , an article of New York State Law Taylor (crater) , on
572-404: A method by Newton, Gregory never described how he obtained these series, and it can only be inferred that he understood the general method by examining scratch work he had scribbled on the back of another letter from 1671. In 1691–1692, Isaac Newton wrote down an explicit statement of the Taylor and Maclaurin series in an unpublished version of his work De Quadratura Curvarum . However, this work
616-478: A philosophical resolution of the paradox, but the mathematical content was apparently unresolved until taken up by Archimedes , as it had been prior to Aristotle by the Presocratic Atomist Democritus . It was through Archimedes's method of exhaustion that an infinite number of progressive subdivisions could be performed to achieve a finite result. Liu Hui independently employed a similar method
660-404: A radius of convergence 0 everywhere. A function cannot be written as a Taylor series centred at 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 . For example, f ( x ) = e can be written as a Laurent series. The generalization of the Taylor series does converge to the value of
704-559: Is infinitely differentiable at x = 0 , and has all derivatives zero there. Consequently, the Taylor series of f ( x ) about x = 0 is identically zero. However, f ( x ) is not the zero function, so does not equal its Taylor series around the origin. Thus, f ( x ) is an example of a non-analytic smooth function . In real analysis , this example shows that there are infinitely differentiable functions f ( x ) whose Taylor series are not equal to f ( x ) even if they converge. By contrast,
748-478: Is formally similar to the Newton series . When the function f is analytic at a , the terms in the series converge to the terms of the Taylor series, and in this sense generalizes the usual Taylor series. In general, for any infinite sequence a i , the following power series identity holds: ∑ n = 0 ∞ u n n ! Δ n
792-401: Is no more than | x | / 9! . For a full cycle centered at the origin ( −π < x < π ) the error is less than 0.08215. 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 ln(1 + x ) and some of its Taylor polynomials around a = 0 . These approximations converge to
836-485: Is the point where the derivatives are considered, after Colin Maclaurin , who made extensive use of this special case of Taylor series in the 18th century. The partial sum formed by the first n + 1 terms of a Taylor series is a polynomial of degree n that is called the n th Taylor polynomial of the function. Taylor polynomials are approximations of a function, which become generally more accurate as n increases. Taylor's theorem gives quantitative estimates on
880-650: Is the polynomial itself. The Maclaurin series of 1 / 1 − x is the geometric series 1 + x + x 2 + x 3 + ⋯ . {\displaystyle 1+x+x^{2}+x^{3}+\cdots .} So, by substituting x for 1 − x , the Taylor series of 1 / x at a = 1 is 1 − ( x − 1 ) + ( x − 1 ) 2 − ( x − 1 ) 3 + ⋯ . {\displaystyle 1-(x-1)+(x-1)^{2}-(x-1)^{3}+\cdots .} By integrating
924-404: Is undefined at 0. More generally, 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, a consequence of Borel's lemma . As a result, the radius of convergence of a Taylor series can be zero. There are even infinitely differentiable functions defined on the real line whose Taylor series have
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#1732837989489968-448: The Taylor series or Taylor expansion of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor series are equal near this point. Taylor series are named after Brook Taylor , who introduced them in 1715. A Taylor series is also called a Maclaurin series when 0
1012-414: The complex plane ) containing x . This implies that the function is analytic at every point of the interval (or disk). The Taylor series of a real or complex-valued function f ( x ) , that is infinitely differentiable at a real or complex number a , is the power series f ( a ) + f ′ ( a ) 1 ! ( x −
1056-1025: The exponential function e is ∑ n = 0 ∞ x n n ! = x 0 0 ! + x 1 1 ! + x 2 2 ! + x 3 3 ! + x 4 4 ! + x 5 5 ! + ⋯ = 1 + x + x 2 2 + x 3 6 + x 4 24 + x 5 120 + ⋯ . {\displaystyle {\begin{aligned}\sum _{n=0}^{\infty }{\frac {x^{n}}{n!}}&={\frac {x^{0}}{0!}}+{\frac {x^{1}}{1!}}+{\frac {x^{2}}{2!}}+{\frac {x^{3}}{3!}}+{\frac {x^{4}}{4!}}+{\frac {x^{5}}{5!}}+\cdots \\&=1+x+{\frac {x^{2}}{2}}+{\frac {x^{3}}{6}}+{\frac {x^{4}}{24}}+{\frac {x^{5}}{120}}+\cdots .\end{aligned}}} The above expansion holds because
1100-425: The holomorphic functions studied in complex analysis always possess a convergent Taylor series, and even the Taylor series of meromorphic functions , which might have singularities, never converge to a value different from the function itself. The complex function e , however, does not approach 0 when z approaches 0 along the imaginary axis, so it is not continuous in the complex plane and its Taylor series
1144-399: The square root , the logarithm , the trigonometric function tangent, and its inverse, arctan . For these functions the Taylor series do not converge if x is far from b . That is, the Taylor series diverges at x if the distance between x and b is larger than the radius of convergence . The Taylor series can be used to calculate the value of an entire function at every point, if
1188-532: The Moon "Taylor" (song) by Jack Johnson, 2004 USS Taylor , the name of several American ships See also [ edit ] All pages with titles beginning with Taylor All pages with titles containing Taylor Tailor (disambiguation) Taylorism , a theory of scientific management, named after Frederick Winslow Taylor Tylor Topics referred to by the same term [REDACTED] This disambiguation page lists articles associated with
1232-485: The Moon "Taylor" (song) by Jack Johnson, 2004 USS Taylor , the name of several American ships See also [ edit ] All pages with titles beginning with Taylor All pages with titles containing Taylor Tailor (disambiguation) Taylorism , a theory of scientific management, named after Frederick Winslow Taylor Tylor Topics referred to by the same term [REDACTED] This disambiguation page lists articles associated with
1276-482: The Taylor series. Thus a function is analytic in an open disk centered at b if and only if its Taylor series converges to the value of the function at each point of the disk. If f ( x ) is equal to the sum of its Taylor series for all x in the complex plane, it is called entire . The polynomials, exponential function e , and the trigonometric functions sine and cosine, are examples of entire functions. Examples of functions that are not entire include
1320-492: The above Maclaurin series, we find the Maclaurin series of ln(1 − x ) , where ln denotes the natural logarithm : − x − 1 2 x 2 − 1 3 x 3 − 1 4 x 4 − ⋯ . {\displaystyle -x-{\tfrac {1}{2}}x^{2}-{\tfrac {1}{3}}x^{3}-{\tfrac {1}{4}}x^{4}-\cdots .} The corresponding Taylor series of ln x at
1364-449: The corresponding Taylor series of ln x at an arbitrary nonzero point a is: ln a + 1 a ( x − a ) − 1 a 2 ( x − a ) 2 2 + ⋯ . {\displaystyle \ln a+{\frac {1}{a}}(x-a)-{\frac {1}{a^{2}}}{\frac {\left(x-a\right)^{2}}{2}}+\cdots .} The Maclaurin series of
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1408-431: The derivative of e with respect to x is also e , and e equals 1. This leaves the terms ( x − 0) in the numerator and n ! in the denominator of each term in the infinite sum. The ancient Greek philosopher Zeno of Elea 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 . Later, Aristotle proposed
1452-412: The error introduced by the use of such approximations. If the Taylor series of a function is convergent , its sum is the limit of the infinite sequence of the Taylor polynomials. A function may differ from the sum of its Taylor series, even if its Taylor series is convergent. A function is analytic at a point x if it is equal to the sum of its Taylor series in some open interval (or open disk in
1496-578: The following two centuries his followers developed further series expansions and rational approximations. In late 1670, James Gregory was shown in a letter from John Collins several Maclaurin series ( sin x , {\textstyle \sin x,} cos x , {\textstyle \cos x,} arcsin x , {\textstyle \arcsin x,} and x cot x {\textstyle x\cot x} ) derived by Isaac Newton , and told that Newton had developed
1540-1712: The 💕 [REDACTED] Look up Taylor in Wiktionary, the free dictionary. Taylor , Taylors or Taylor's may refer to: People [ edit ] Taylor (surname) List of people with surname Taylor Taylor (given name) , including Tayla and Taylah Taylor sept , a branch of Scottish clan Cameron Justice Taylor (disambiguation) Places [ edit ] Australia [ edit ] Electoral district of Taylor , South Australia Taylor, Australian Capital Territory , planned suburb Canada [ edit ] Taylor, British Columbia United States [ edit ] Taylor, Alabama Taylor, Arizona Taylor, Arkansas Taylor, Indiana Taylor, Louisiana Taylor, Maryland Taylor, Michigan Taylor, Mississippi Taylor, Missouri Taylor, Nebraska Taylor, North Dakota Taylor, New York Taylor, Beckham County, Oklahoma Taylor, Cotton County, Oklahoma Taylor, Pennsylvania Taylors, South Carolina Taylor, Texas Taylor, Utah Taylor, Washington Taylor, West Virginia Taylor, Wisconsin Taylor, Wyoming Taylor County (disambiguation) Taylor Township (disambiguation) Businesses and organisations [ edit ] Taylor's (department store) in Quebec, Canada Taylor Guitars , an American guitar manufacturer Taylor University , in Upland, Indiana, U.S. Taylor's University , commonly referred to as Taylor's, in Subang Jaya, Selangor, Malaysia Taylor's College John Taylor & Co , or Taylor's Bell Foundry, Taylor's of Loughborough, or Taylor's, in England Taylor Company ,
1584-431: The function itself for any bounded continuous function on (0,∞) , and this can be done by using the calculus of finite differences . Specifically, the following theorem, due to Einar Hille , that for any t > 0 , lim h → 0 + ∑ n = 0 ∞ t n n ! Δ h n f (
1628-399: 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 error incurred in approximating a function by its n th-degree Taylor polynomial is called the remainder or residual and is denoted by the function R n ( x ) . Taylor's theorem can be used to obtain a bound on the size of
1672-519: The inverse Gudermannian function ), arcsec ( 2 e x ) , {\textstyle \operatorname {arcsec} {\bigl (}{\sqrt {2}}e^{x}{\bigr )},} and 2 arctan e x − 1 2 π {\textstyle 2\arctan e^{x}-{\tfrac {1}{2}}\pi } (the Gudermannian function). However, thinking that he had merely redeveloped
1716-744: The remainder . In general, Taylor series need not be convergent at all. In fact, the set of functions with a convergent Taylor series is a meager set in the Fréchet space of smooth functions . Even if the Taylor series of a function f does converge, its limit need not be equal to the value of the function f ( x ) . For example, the function f ( x ) = { e − 1 / x 2 if x ≠ 0 0 if x = 0 {\displaystyle f(x)={\begin{cases}e^{-1/x^{2}}&{\text{if }}x\neq 0\\[3mu]0&{\text{if }}x=0\end{cases}}}
1760-409: The special case of the Taylor result in the mid-18th century. If f ( x ) is given by a convergent power series in an open disk centred at b in the complex plane (or an interval in the real line), it is said to be analytic in this region. Thus for x in this region, f is given by a convergent power series f ( x ) = ∑ n = 0 ∞
1804-491: The title Taylor . If an internal link led you here, you may wish to change the link to point directly to the intended article. Retrieved from " https://en.wikipedia.org/w/index.php?title=Taylor&oldid=1220183788 " Categories : Disambiguation pages Place name disambiguation pages Hidden categories: Short description is different from Wikidata All article disambiguation pages All disambiguation pages Taylor From Misplaced Pages,
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1848-499: The title Taylor . If an internal link led you here, you may wish to change the link to point directly to the intended article. Retrieved from " https://en.wikipedia.org/w/index.php?title=Taylor&oldid=1220183788 " Categories : Disambiguation pages Place name disambiguation pages Hidden categories: Short description is different from Wikidata All article disambiguation pages All disambiguation pages Taylor Series In mathematics ,
1892-636: The value of the function, and of all of its derivatives, are known at a single point. Uses of the Taylor series for analytic functions include: Pictured is an accurate approximation of sin x around the point x = 0 . The pink curve is a polynomial of degree seven: sin x ≈ x − x 3 3 ! + x 5 5 ! − x 7 7 ! . {\displaystyle \sin {x}\approx x-{\frac {x^{3}}{3!}}+{\frac {x^{5}}{5!}}-{\frac {x^{7}}{7!}}.\!} The error in this approximation
1936-438: Was never completed and the relevant sections were omitted from the portions published in 1704 under the title Tractatus de Quadratura Curvarum . It was not until 1715 that a general method for constructing these series for all functions for which they exist was finally published by Brook Taylor , after whom the series are now named. The Maclaurin series was named after Colin Maclaurin , a Scottish mathematician, who published
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