Taylor's Series

Taylor’s Series 

(or Taylor series), a power series of the form

where f(x) is a function having derivatives of all orders at x = a.

In many cases of practical importance, Taylor’s series converges to f(x) on some interval with center at a:

(this formula was published in 1715 by B. Taylor). The difference R_n(x) = f(x) – S_n(x), where S_n(x) is the sum of the first n + 1 terms of series (1), is called the remainder. Formula (2) is valid if R_n(x)→ 0 as n→ ∞. A Taylor’s series can be represented in the form

which is also applicable to functions of several variables.

When a = 0, the expansion of a function in a Taylor’s series assumes the form

Such a series has been traditionally, although incorrectly, called a Maclaurin’s series. Examples of Maclaurin’s series are

Series (3) is a generalization of Newton’s binomial formula to the case where the exponents may be fractional or negative. This series converges for –1 < x < 1 if m < –1, for –1 < x ≤ 1 if – 1 < m < 0, and for – 1 ≤ x ≤ 1 if m > 0. Series (4), (5), and (6) converge for any value of x. Series (7) converges for –1 < x ≤ 1.

If a function f (z) of a complex variable z is regular at the point a, the function can be expanded in a Taylor’s series in powers of z – a within a circle having a center at a and a radius equal to the distance from a to the nearest singular point of f (z). Outside this circle, the Taylor’s series diverges; its behavior on the circumference of the circle of convergence may be very complicated. The radius of the circle of convergence is expressed in terms of the coefficients of the Taylor’s series (seeRADIUS OF CONVERGENCE).

Taylor’s series are used in the study of functions and in approximate calculations (see also).

REFERENCES

Khinchin, A. Ia. Kratkii kurs matematicheskogo analiza. Moscow, 1953.
Fikhtengol’ts, G. M. Kurs differentsial’nogo i integral’nogo ischisleniia, 7th ed., vol. 2. Moscow, 1969.