Taylor series

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 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 better 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 (or open disk in the complex plane) containing . This implies that the function is analytic at every point of the interval (or disk).

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
:$f(a)+\frac (x-a)+ \frac (x-a)^2+\frac(x-a)^3+ \cdots,$

where denotes the factorial of . In the more compact sigma notation, this can be written as
:$\sum_ ^ \frac (x-a)^,$
where 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.)

When , the series is also called a Maclaurin series.

Examples

The Taylor series of any polynomial is the polynomial itself.

The Maclaurin series of is the geometric series

:$1 + x + x^2 + x^3 + \cdots.$

So, by substituting for , the Taylor series of at is

:$1 - (x-1) + (x-1)^2 - (x-1)^3 + \cdots.$

By integrating the above Maclaurin series, we find the Maclaurin series of , where denotes the natural logarithm:

:$-x - \tfracx^2 - \tfracx^3 - \tfracx^4 - \cdots.$

The corresponding Taylor series of at is

:$(x-1) - \tfrac(x-1)^2 + \tfrac(x-1)^3 - \tfrac(x-1)^4 + \cdots,$

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

:$\ln a + \frac (x - a) - \frac\frac + \cdots.$

The Maclaurin series of the exponential function is

:$\begin \sum_^\infty \frac &= \frac + \frac + \frac + \frac + \frac + \frac+ \cdots \\ &= 1 + x + \frac + \frac + \frac + \frac + \cdots. \end$

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 the use of Taylor series and closely related methods were given by Madhava of Sangamagrama. Though no record of his work survives, writings of later Indian mathematicians suggest that he found a number of special cases of the Taylor series, including those for the trigonometric functions of sine, cosine, tangent, and arctangent. Madhava founded the Kerala school of astronomy and mathematics, and during the following two centuries its scholars developed further series expansions and rational approximations.

In the 17th century, James Gregory also worked in this area and published several Maclaurin series. It was not until 1715 however that a general method for constructing these series for all functions for which they exist was finally provided by Brook Taylor, 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

If is given by a convergent power series in an open disk centred at in the complex plane (or an interval in the real line), it is said to be analytic in this region. Thus for in this region, is given by a convergent power series
:$f(x) = \sum_^\infty a_n(x-b)^n.$
Differentiating by the above formula times, then setting gives:
:$\frac = a_n$
and so the power series expansion agrees with the Taylor series. Thus a function is analytic in an open disk centred 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 function