As the degree of the Taylor polynomial rises, it approaches the correct function. This image showssinx and its Taylor approximations by polynomials of degree1,3,5,7,9,11, and13 atx = 0.
Inmathematics, theTaylor series orTaylor expansion of afunction is aninfinite sum of terms that are expressed in terms of the function'sderivatives 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 afterBrook Taylor, who introduced them in 1715. A Taylor series is also called aMaclaurin series when 0 is the point where the derivatives are considered, afterColin Maclaurin, who made extensive use of this special case of Taylor series in the 18th century.
Thepartial sum formed by the firstn + 1 terms of a Taylor series is apolynomial of degreen that is called thenthTaylor polynomial of the function. Taylor polynomials are approximations of a function, which become generally more accurate asn increases.Taylor's theorem gives quantitative estimates on the error introduced by the use of such approximations. If the Taylor series of a function isconvergent, its sum is thelimit of theinfinite 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 isanalytic at a pointx if it is equal to the sum of its Taylor series in someopen interval (oropen disk in thecomplex plane) containingx. This implies that the function is analytic at every point of the interval (or disk).
The above expansion holds because the derivative ofex with respect tox is alsoex, ande0 equals 1. This leaves the terms(x − 0)n in the numerator andn! in the denominator of each term in the infinite sum.
Theancient Greek philosopherZeno of Elea considered the problem of summing an infinite series to achieve a finite result, but rejected it as an impossibility;[3] the result wasZeno's paradox. Later,Aristotle proposed a philosophical resolution of the paradox, but the mathematical content was apparently unresolved until taken up byArchimedes, as it had been prior to Aristotle by the Presocratic AtomistDemocritus. It was through Archimedes'smethod of exhaustion that an infinite number of progressive subdivisions could be performed to achieve a finite result.[4]Liu Hui independently employed a similar method a few centuries later.[5]
In the 14th century, the earliest examples of specific Taylor series (but not the general method) were given by Indian mathematicianMadhava of Sangamagrama.[6] Though no record of his work survives, writings of his followers in theKerala school of astronomy and mathematics suggest that he found the Taylor series for thetrigonometric functions ofsine,cosine, andarctangent (seeMadhava series). During the following two centuries his followers developed further series expansions and rational approximations.
In late 1670,James Gregory was shown in a letter fromJohn Collins several Maclaurin series( and) derived byIsaac Newton, and told that Newton had developed 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 (the integral of), (theintegral ofsec, the inverseGudermannian function), and (the Gudermannian function). However, thinking that he had merely redeveloped 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.[7]
In 1691–1692, Isaac Newton wrote down an explicit statement of the Taylor and Maclaurin series in an unpublished version of his workDe Quadratura Curvarum. However, this work was never completed and the relevant sections were omitted from the portions published in 1704 under the titleTractatus 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 byBrook Taylor,[8] after whom the series are now named.
The Maclaurin series was named afterColin Maclaurin, a Scottish mathematician, who published a special case of the Taylor result in the mid-18th century.
The functione(−1/x2) is not analytic atx = 0: the Taylor series is identically 0, although the function is not.
Iff (x) is given by a convergent power series in an open disk centred atb in the complex plane (or an interval in the real line), it is said to beanalytic in this region. Thus forx in this region,f is given by a convergent power series
Differentiating byx the above formulan times, then settingx =b gives:
and so the power series expansion agrees with the Taylor series. Thus a function is analytic in an open disk centered atb if and only if its Taylor series converges to the value of the function at each point of the disk.
Iff (x) is equal to the sum of its Taylor series for allx in the complex plane, it is calledentire. The polynomials,exponential functionex, and thetrigonometric functions sine and cosine, are examples of entire functions. Examples of functions that are not entire include thesquare root, thelogarithm, thetrigonometric function tangent, and its inverse,arctan. For these functions the Taylor series do notconverge ifx is far fromb. That is, the Taylor seriesdiverges atx if the distance betweenx andb is larger than theradius of convergence. The Taylor series can be used to calculate the value of an entire function at every point, if 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:
The partial sums (theTaylor polynomials) of the series can be used as approximations of the function. These approximations are good if sufficiently many terms are included.
Differentiation and integration of power series can be performed term by term and is hence particularly easy.
The (truncated) series can be used to compute function values numerically, (often by recasting the polynomial into theChebyshev form and evaluating it with theClenshaw algorithm).
Algebraic operations can be done readily on the power series representation; for instance,Euler's formula follows from Taylor series expansions for trigonometric and exponential functions. This result is of fundamental importance in such fields asharmonic analysis.
Approximations using the first few terms of a Taylor series can make otherwise unsolvable problems possible for a restricted domain; this approach is often used in physics.
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 forln(1 +x) only provide accurate approximations in the range−1 <x ≤ 1. Forx > 1, Taylor polynomials of higher degree provide worse approximations.The Taylor approximations forln(1 +x) (black). Forx > 1, the approximations diverge.
Pictured is an accurate approximation ofsinx around the pointx = 0. The pink curve is a polynomial of degree seven:
The error in this approximation is no more than|x|9 / 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 functionln(1 +x) and some of its Taylor polynomials arounda = 0. These approximations converge to the function only in the region−1 <x ≤ 1; outside of this region the higher-degree Taylor polynomials areworse approximations for the function.
Theerror incurred in approximating a function by itsnth-degree Taylor polynomial is called theremainder orresidual and is denoted by the functionRn(x). Taylor's theorem can be used to obtain a bound on thesize of the remainder.
In general, Taylor series need not beconvergent at all. In fact, the set of functions with a convergent Taylor series is ameager set in theFréchet space ofsmooth functions. Even if the Taylor series of a functionf does converge, its limit need not be equal to the value of the functionf (x). For example, the function
isinfinitely differentiable atx = 0, and has all derivatives zero there. Consequently, the Taylor series off (x) aboutx = 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 anon-analytic smooth function.
Inreal analysis, this example shows that there areinfinitely differentiable functionsf (x) whose Taylor series arenot equal tof (x) even if they converge. By contrast, theholomorphic functions studied incomplex analysis always possess a convergent Taylor series, and even the Taylor series ofmeromorphic functions, which might have singularities, never converge to a value different from the function itself. The complex functione−1/z2, however, does not approach 0 whenz approaches 0 along the imaginary axis, so it is notcontinuous in the complex plane and its Taylor series is undefined at 0.
More generally, every sequence of real or complex numbers can appear ascoefficients in the Taylor series of an infinitely differentiable function defined on the real line, a consequence ofBorel's lemma. As a result, theradius of convergence of a Taylor series can be zero. There are even infinitely differentiable functions defined on the real line whose Taylor series have a radius of convergence 0 everywhere.[9]
A function cannot be written as a Taylor series centred at asingularity; in these cases, one can often still achieve a series expansion if one allows also negative powers of the variablex; seeLaurent series. For example,f (x) =e−1/x2 can be written as a Laurent series.
The generalization of the Taylor series does converge to the value of the function itself for anyboundedcontinuous function on(0,∞), and this can be done by using the calculus offinite differences. Specifically, the following theorem, due toEinar Hille, that for anyt > 0,[10]
HereΔn h is thenth finite difference operator with step sizeh. The series is precisely the Taylor series, except that divided differences appear in place of differentiation: the series is formally similar to theNewton series. When the functionf is analytic ata, 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 sequenceai, the following power series identity holds:
The last series is known asMercator series, named afterNicholas Mercator (since it was published in his 1668 treatiseLogarithmotechnia).[14] Both of these series converge for. (In addition, the series forln(1 −x) converges forx = −1, and the series forln(1 +x) converges forx = 1.)[13]
(Ifn = 0, this product is anempty product and has value 1.) It converges for for any real or complex numberα.
Whenα = −1, this is essentially the infinite geometric series mentioned in the previous section. The special casesα =1/2 andα = −1/2 give thesquare root function and itsinverse:[16]
Several methods exist for the calculation of Taylor series of a large number of functions. One can attempt to use the definition of the Taylor series, though this often requires generalizing the form of the coefficients according to a readily apparent pattern. Alternatively, 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. In some cases, one can also derive the Taylor series by repeatedly applyingintegration by parts. Particularly convenient is the use ofcomputer algebra systems to calculate Taylor series.
In order to compute the 7th degree Maclaurin polynomial for the function
one may first rewrite the function as
the composition of two functions and The Taylor series for the natural logarithm is (usingbig O notation)
and for the cosine function
The first several terms from the second series can be substituted into each term of the first series. Because the first term in the second series has degree 2, three terms of the first series suffice to give a 7th-degree polynomial:
Since the cosine is aneven function, the coefficients for all the odd powers are zero.
Here we employ a method called "indirect expansion" to expand the given function. This method uses the known Taylor expansion of the exponential function. In order to expand(1 +x)ex as a Taylor series inx, we use the known Taylor series of functionex:
Classically,algebraic functions are defined by an algebraic equation, andtranscendental functions (including those discussed above) are defined by some property that holds for them, such as adifferential equation. For example, theexponential 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 ananalytic function by its Taylor series.
Taylor series are used to define functions and "operators" in diverse areas of mathematics. In particular, this is true in areas where the classical definitions of functions break down. For example, using Taylor series, one may extend analytic functions to sets of matrices and operators, such as thematrix exponential ormatrix logarithm.
In other areas, such as formal analysis, it is more convenient to work directly with thepower series themselves. Thus one may define a solution of a differential equationas a power series which, one hopes to prove, is the Taylor series of the desired solution.
which is to be understood as a still more abbreviatedmulti-index version of the first equation of this paragraph, with a full analogy to the single variable case.
The trigonometricFourier series enables one to express aperiodic function (or a function defined on a closed interval[a,b]) as an infinite sum oftrigonometric functions (sines andcosines). In this sense, the Fourier series is analogous to Taylor series, since the latter allows one to express a function as an infinite sum ofpowers. Nevertheless, the two series differ from each other in several relevant issues:
The finite truncations of the Taylor series off (x) about the pointx =a are all exactly equal tof ata. In contrast, the Fourier series is computed by integrating over an entire interval, so there is generally no such point where all the finite truncations of the series are exact.
The computation of Taylor series requires the knowledge of the function on an arbitrary smallneighbourhood of a point, whereas the computation of the Fourier series requires knowing the function on its whole domaininterval. In a certain sense one could say that the Taylor series is "local" and the Fourier series is "global".
The Taylor series is defined for a function which has infinitely many derivatives at a single point, whereas the Fourier series is defined for anyintegrable function. In particular, the function could be nowhere differentiable. (For example,f (x) could be aWeierstrass function.)
The convergence of both series has very different properties. Even if the Taylor series has positive convergence radius, the resulting series may not coincide with the function; but if the function is analytic then the series convergespointwise to the function, anduniformly on every compact subset of the convergence interval. Concerning the Fourier series, if the function issquare-integrable then the series converges inquadratic mean, but additional requirements are needed to ensure the pointwise or uniform convergence (for instance, if the function is periodic and of class C1 then the convergence is uniform).
Finally, in practice one wants to approximate the function with a finite number of terms, say with a Taylor polynomial or a partial sum of the trigonometric series, respectively. In the case of the Taylor series the error is very small in a neighbourhood of the point where it is computed, while it may be very large at a distant point. In the case of the Fourier series the error is distributed along the domain of the function.
Feigenbaum, L. (1985). "Brook Taylor and the method of increments".Archive for History of Exact Sciences.34 (1–2):1–140.doi:10.1007/bf00329903.S2CID122105736.
Hille, Einar;Phillips, Ralph S. (1957).Functional analysis and semi-groups. AMS Colloquium Publications. Vol. 31. American Mathematical Society.
Hofmann, Josef Ehrenfried (1939). "On the Discovery of the Logarithmic Series and Its Development in England up to Cotes".National Mathematics Magazine.14 (1):33–45.doi:10.2307/3028095.JSTOR3028095.
Lindberg, David (2007).The Beginnings of Western Science (2nd ed.). University of Chicago Press.ISBN978-0-226-48205-7.
Malet, Antoni (1993). "James Gregorie on Tangents and the "Taylor" Rule for Series Expansions".Archive for History of Exact Sciences.46 (2):97–137.doi:10.1007/BF00375656.JSTOR41133959.S2CID120101519.
