Taylor series

Taylor's theorem gives quantitative estimates on the error introduced by the use of such approximations.

and more generally, the corresponding Taylor series of ln x at an arbitrary nonzero point a is:

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;[3] the result was Zeno's paradox.

It was through Archimedes's method 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 mathematician Madhava of Sangamagrama.

[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).

In late 1670, James Gregory was shown in a letter from John Collins several Maclaurin series (

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[7] In 1691–1692, Isaac Newton wrote down an explicit statement of the Taylor and Maclaurin series in an unpublished version of his work De Quadratura Curvarum.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.

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.

If f (x) is equal to the sum of its Taylor series for all x in the complex plane, it is called entire.

That is, the Taylor series diverges at x if the distance between x and b is larger than the radius of convergence.

Uses of the Taylor series for analytic functions include: Pictured is an accurate approximation of sin x around the point x = 0.

In contrast, also shown is a picture of the natural logarithm function ln(1 + x) and some of its Taylor polynomials around a = 0.

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.

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.

There are even infinitely differentiable functions defined on the real line whose Taylor series have a radius of convergence 0 everywhere.

In general, for any infinite sequence ai, the following power series identity holds:

When α = −1, this is essentially the infinite geometric series mentioned in the previous section.

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.

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.

Taylor series are used to define functions and "operators" in diverse areas of mathematics.

In other areas, such as formal analysis, it is more convenient to work directly with the power series themselves.

A second-order Taylor series expansion of a scalar-valued function of more than one variable can be written compactly as

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.

In order to compute a second-order Taylor series expansion around point (a, b) = (0, 0) of the function

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 .
The function e (−1/ x 2 ) is not analytic at x = 0 : the Taylor series is identically 0, although the function is not.
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.
The exponential function e x (in blue), and the sum of the first n + 1 terms of its Taylor series at 0 (in red).
Second-order Taylor series approximation (in orange) of a function f ( x , y ) = e x ln(1 + y ) around the origin.