Taylor series


In mathematical analysis, 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 18th century.
The partial sum formed by the first terms of a Taylor series is a polynomial of degree that is called the th Taylor polynomial of the function. Taylor polynomials are approximations of a function, which become generally more accurate as 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 if it is equal to the sum of its Taylor series in some open interval containing. This implies that the function is analytic at every point of the interval.

Definition

The Taylor series of a real or complex-valued function, that is infinitely differentiable at a real or complex number, is the power series
Here, denotes the factorial of. The function denotes the th derivative of evaluated at the point. The derivative of order zero of is defined to be itself and and are both defined to be 1. This series can be written by using sigma notation, as in the right side formula. With, the Maclaurin series takes the form:

Examples

The Taylor series of any polynomial is the polynomial itself.
The Maclaurin series of is the geometric series
So, by substituting for, the Taylor series of at is
By integrating the above Maclaurin series, we find the Maclaurin series of, where denotes the natural logarithm:
The corresponding Taylor series of at is
and more generally, the corresponding Taylor series of at an arbitrary nonzero point is
The Maclaurin series of the exponential function is
The above expansion holds because the derivative of with respect to is also, and equals 1. This leaves the terms in the numerator and in the denominator of each term in the infinite sum.

History

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 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 a few centuries later.
In the 14th century, the earliest examples of specific Taylor series were given by the 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 the following two centuries, his followers developed further series expansions and rational approximations.
The Tantrasangraha-vakhya ''from the Kerala school gives the series in verse, which when translated to mathematical notation, can be written as:

where, for the series reduce to the standard power series for these trigonometric functions, for example:
In late 1670, James Gregory was shown in a letter from John Collins several Maclaurin series and 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 , and . 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.
In 1691–1692, Newton wrote down an explicit statement of the Taylor and Maclaurin series in an unpublished version of his work De Quadratura Curvarum. It was the earliest explicit formulation of the general Taylor series. However, this work by Newton 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 a special case of the Taylor result in the mid-18th century.

Analytic functions

If is given by a convergent power series in an open disk centred at in the complex plane, it is said to be analytic in this region. Thus for in this region, is given by a convergent power series
Differentiating by the above formula times, then setting gives
and so the power series expansion agrees with the Taylor series. Thus, a function is analytic in an open disk centered at if and only if its Taylor series converges to the value of the function at each point of the disk.
If is equal to the sum of its Taylor series for all in the complex plane, it is called entire. The polynomials, exponential function, and the trigonometric functions of 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 is far from. That is, the Taylor series diverges at if the distance between and is larger than the radius of convergence. The Taylor series can be used to calculate the value of an entire function at every point, provided the value of the function and all its derivatives are known at a single point.
Uses of the Taylor series for analytic functions include:
  • The partial sums 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 are hence particularly easy.
  • An analytic function is uniquely extended to a holomorphic function on an open disk in the complex plane. This makes the machinery of complex analysis available.
  • The 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.

    Approximation error and convergence

The error incurred in approximating a function by its th-degree Taylor polynomial is called the remainder and is denoted by the function. 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. 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 does converge, its limit need not be equal to the value of the function. For example, the function
is infinitely differentiable at, and has all derivatives zero there. Consequently, the Taylor series of about is identically zero. However, is not the zero function, so it does not equal its Taylor series around the origin. Thus, is an example of a non-analytic smooth function. This example shows that there are infinitely differentiable functions in real analysis, whose Taylor series are not equal to 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 a meromorphic function, which might have singularities, never converges to a value different from the function itself. The complex function, however, does not approach 0 when approaches 0 along the imaginary axis, so it is not continuous in the complex plane and its Taylor series is undefined at 0.
Every sequence of real or complex numbers can appear more generally 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.
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 also allows negative powers of the variable. For example, can be written as a Laurent series.