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 (x – a) of degree n and a remainder Rx(x). In the neighborhood of a, Rn (x) is an infinitesimal of higher order than (x – a)n—that is, Rn(x) = αn(x) (x – a)n, where αn(x) → 0 as x → a.
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.
REFERENCESKhinchin, 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.