The Taylor series for a function f(x) with center at point x0 having at this point derivatives of all orders, is defined, for the vicinity of x0, as follows:
For x0 = 0 Taylor series is called Maclaurin series.
For the Taylor series (as for any power series) there exists a real number r(0 ≤ r ≤ ∞) called the radius of convergence, such that for |x−x0| > r, series (1) converges absolutely, and for |x−x0| > r it diverges. Within the circle of convergence (i.e., in a circle |x−x0| ≤ q < r, where q is any real number) Taylor series (1) converges uniformly. Convergent power series can be added and multiplied term by term.
Taylor power series can be differentiated and integrated. Series obtained as a result of termwise differentiation or termwise integration from 0 to x have the same radius of convergence as the initial series. This property of Taylor series is often used in solving problems of hydrodynamics and heat transfer, for solving differential equations and in integration of complex transcendental functions.
If f(z) is an analytic function of a complex variable inside a circle with center at point x0, then it can be expanded into a Taylor series within this circle. A derivative of kth order in this case is represented in integral form
where Г is a circle with center x0 laying inside this circle. The uniqueness of the expansion is associated with the fact that in this case any power series is a Taylor series for its sum.
If x0 is a real number and f is defined in the vicinity of x0 by a set of real numbers and has derivatives of all orders at x0, then the function f cannot in the vicinity of x0 be the sum of its own Taylor series. One and the same power series can be a Taylor series for different real functions.
A sufficient condition for Taylor series convergence to a real function on the interval |x−x0| < r is in total the restriction of all its derivatives in this interval (i.e., |f(k)(x)<L|, where L is independent of k for |x−x0| < r).
A real-valued function which has n derivatives at x0 is often represented in the form of a partial sum of n – 1 terms of a Taylor series (Taylor polynomial of degree n – 1) and a remainder terra (Taylor formula):
The remainder term Rn(x) is
By applying the mean value theorem, the remainder term can be written as a Lagrange formula
If the function f(x) has all derivatives in the interval |x−x0| < r and limn→∞ Rn(x), then
and a series converges uniformly to f(x) on any interval |x−x0| ≤ q < r.
The Taylor formula generalizes to the case of a function of several variables. If f(x1, ..., xn) is a real function, which has all continuous derivatives of order ≤ m in a certain vicinity D of the point x0, then
The remainder term Rm(x1, ..., xn) satisfies the relation similar to relations for a function of one variable. If a function f(x1, ..., xn) has in D all continuous partial derivatives and limm→∞Rm(x1, ..., xn) in D, then the Taylor formula brings about an expansion into a multiple series. A function, which can be expanded into a power series, convergent in a certain vicinity of the point (x01, ..., x0n) is called analytic in this vicinity.
The Taylor formula allows us to reduce the study of the properties of a differentiable function to a simpler problem of studying the properties of a corresponding Taylor series and of evaluating the remainder term. The application of the Taylor formula for calculation of convergence of series and integrals, and for estimation of the rate of their convergence and divergence is based on this property.