As the degree of the Taylor polynomial rises, it approaches the correct function. This image shows sin x and its Taylor approximations by polynomials of degree 1, 3, 5, 7, 9, 11, and 13 at x = 0.
In
mathematics, 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 is the point where the derivatives are considered, after
Colin Maclaurin, who made extensive use of this special case of Taylor series in the mid-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 nth Taylor polynomial of the function. Taylor polynomials are approximations of a function, which become generally better as n increases.
Taylor's theorem gives quantitative estimates on 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 the
complex plane) containing x. This implies that the function is analytic at every point of the interval (or disk).
where f^{(n)}(a) denotes the nth
derivative of f evaluated at the point a. (The derivative of order zero of f is defined to be f itself and (x − a)^{0} and 0!are both defined to be 1.)
When a = 0, the series is also called a Maclaurin series.^{
[1]}
Examples
The Taylor series of any
polynomial is the polynomial itself.
The above expansion holds because the derivative of e^{x} with respect to x is also e^{x}, and e^{0} equals 1. This leaves the terms (x − 0)^{n} in the numerator and n! in the denominator of each term in the infinite sum.
History
The
ancient Greek philosopherZeno of Elea considered the problem of summing an infinite series to achieve a finite result, but rejected it as an impossibility;^{
[2]} the result was
Zeno's paradox. Later,
Aristotle proposed 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.^{
[3]}Liu Hui independently employed a similar method a few centuries later.^{
[4]}
In the 14th century, the earliest examples of specific Taylor series (but not the general method) were given by
Madhava of Sangamagrama.^{
[5]}^{
[6]} 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 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 (${\textstyle \sin x,}$${\textstyle \cos x,}$${\textstyle \arcsin x,}$ and ${\textstyle x\cot x}$) derived by
Isaac 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 ${\textstyle \arctan x,}$${\textstyle \tan x,}$${\textstyle \sec x,}$${\textstyle \ln \,\sec x}$ (the integral of $\tan$),${\textstyle \ln \,\tan {\tfrac {1}{2}}{{\bigl (}{\tfrac {1}{2}}\pi +x{\bigr )}}}$ (the
integral of sec, the inverse
Gudermannian function), ${\textstyle \operatorname {arcsec} {\bigl (}{\sqrt {2}}e^{x}{\bigr )},}$ and ${\textstyle 2\arctan e^{x}-{\tfrac {1}{2}}\pi }$ (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 work De Quadratura Curvarum. However, this work 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,^{
[8]} after whom the series are now named.
The Maclaurin series was named after
Colin Maclaurin, a professor in Edinburgh, who published the special case of the Taylor result in the mid-18th century.
Analytic functions
The function e^{(−1/x2)} is not analytic at x = 0: the Taylor series is identically 0, although the function is not.
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
$f(x)=\sum _{n=0}^{\infty }a_{n}(x-b)^{n}.$
Differentiating by x the above formula n times, then setting x = b gives:
${\frac {f^{(n)}(b)}{n!}}=a_{n}$
and so the power series expansion agrees with 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 functione^{x}, and the
trigonometric functions sine and cosine, are examples of entire functions. Examples of functions that are not entire include 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 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 (the
Taylor 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 the
Chebyshev form and evaluating it with the
Clenshaw 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 as
harmonic 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 for ln(1 + x) only provide accurate approximations in the range −1 < x ≤ 1. For x > 1, Taylor polynomials of higher degree provide worse approximations.
The Taylor approximations for ln(1 + x) (black). For x > 1, the approximations diverge.
Pictured is an accurate approximation of sin x around the point x = 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 function ln(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 error incurred in approximating a function by its nth-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 the remainder.
In general, Taylor series need not be
convergent at all. And in fact the set of functions with a convergent Taylor series is a
meager set in the
Fréchet space of
smooth functions. And even if the Taylor series of a function f does converge, its limit need not in general be equal to the value of the function f (x). For example, the function
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 functionsf (x) whose Taylor series are not equal to f (x) even if they converge. By contrast, 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^{−1/z2}, 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 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 a radius of convergence 0 everywhere.^{
[9]}
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^{−1/x2} can be written as a Laurent series.
Generalization
There is, however, a generalization^{
[10]}^{
[11]} of the Taylor series that does converge to the value of the function itself for any
boundedcontinuous function on (0,∞), using the calculus of
finite differences. Specifically, one has the following theorem, due to
Einar Hille, that for any t > 0,
Here Δ^{n} _{h} is the nth 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 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:
(If n = 0, this product is an
empty product and has value 1.) It converges for $|x|<1$ for any real or complex number α.
When α = −1, this is essentially the infinite geometric series mentioned in the previous section. The special cases α = 1/2 and α = −1/2 give the
square root function and its
inverse:
All angles are expressed in
radians. The numbers B_{k} appearing in the expansions of tan x are the
Bernoulli numbers. The E_{k} in the expansion of sec x are
Euler numbers.
Hyperbolic functions
The
hyperbolic functions have Maclaurin series closely related to the series for the corresponding trigonometric functions:
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 applying
integration by parts. Particularly convenient is the use of
computer algebra systems to calculate Taylor series.
First example
In order to compute the 7th degree Maclaurin polynomial for the function
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:
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)e^{x} as a Taylor series in x, we use the known Taylor series of function e^{x}:
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. 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 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 the
matrix exponential or
matrix logarithm.
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.
Taylor series in several variables
The Taylor series may also be generalized to functions of more than one variable with^{
[14]}^{
[15]}
where Df (a) is the
gradient of f evaluated at x = a and D^{2}f (a) is the
Hessian matrix. Applying the
multi-index notation the Taylor series for several variables becomes
which is to be understood as a still more abbreviated
multi-index version of the first equation of this paragraph, with a full analogy to the single variable case.
Example
Second-order Taylor series approximation (in orange) of a function f (x,y) = e^{x} ln(1 + y) around the origin.
In order to compute a second-order Taylor series expansion around point (a, b) = (0, 0) of the function
$f(x,y)=e^{x}\ln(1+y),$
one first computes all the necessary partial derivatives:
The trigonometric
Fourier series enables one to express a
periodic function (or a function defined on a closed interval a,b) as an infinite sum of
trigonometric functions (
sines and
cosines). In this sense, the Fourier series is analogous to Taylor series, since the latter allows one to express a function as an infinite sum of
powers. Nevertheless, the two series differ from each other in several relevant issues:
The finite truncations of the Taylor series of f (x) about the point x = a are all exactly equal to f at a. 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 small
neighbourhood of a point, whereas the computation of the Fourier series requires knowing the function on its whole domain
interval. 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 any
integrable function. In particular, the function could be nowhere differentiable. (For example, f (x) could be a
Weierstrass 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 converges
pointwise to the function, and
uniformly on every compact subset of the convergence interval. Concerning the Fourier series, if the function is
square-integrable then the series converges in
quadratic mean, but additional requirements are needed to ensure the pointwise or uniform convergence (for instance, if the function is periodic and of class C^{1} 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.
^Hille, Einar;
Phillips, Ralph S. (1957), Functional analysis and semi-groups, AMS Colloquium Publications, vol. 31, American Mathematical Society, pp. 300–327.