# Taylor's Formula

*The Great Soviet Encyclopedia*(1979). It might be outdated or ideologically biased.

## Taylor’s Formula

the formula

If *f* is a function having an *n*th 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 *R _{x}*(

*x*). In the neighborhood of a,

*R*(

_{n}*x*) is an infinitesimal of higher order than (

*x*–

*a*)

^{n}—that is,

*R*(

_{n}*x*) =

*α*(x) (

_{n}*x*–

*a*)

^{n}, where α

_{n}(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 ξ_{1} 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.