Taylor's Formula

The following article is from The Great Soviet Encyclopedia (1979). It might be outdated or ideologically biased.

Taylor’s Formula

 

the formula

If f is a function having an nth derivative at the point x = a, then Taylor’s formula provides a representation of f as the sum of a polynomial in (xa) of degree n and a remainder Rx(x). In the neighborhood of a, Rn (x) is an infinitesimal of higher order than (xa)n—that is, Rn(x) = αn(x) (xa)n, where αn(x) → 0 as xa.

If the (n + 1)th derivative exists on the interval between a and x, then the remainder may be expressed in Lagrange’s form or in Cauchy’s form. Lagrange’s form is

Cauchy’s form is

Here, ξ, and ξ1 are some points on the given interval.

At the point a, the order of contact of the graph of the polynomial in Taylor’s formula and the graph of f(x) is at least n.

Taylor’s formula is used in the study of functions and in approximate calculations.

REFERENCES

Khinchin, A. Ia. Kratkii kurs matematicheskogo analiza. Moscow, 1953.
Fikhtengol’ts, G. M. Kurs differentsial’nogo i integral’nogo ischisleniia, 7th ed., vol. 1. Moscow, 1969.
The Great Soviet Encyclopedia, 3rd Edition (1970-1979). © 2010 The Gale Group, Inc. All rights reserved.
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