Taylor polynomial

of degree , for a function that is times differentiable at

The polynomial

The values of the Taylor polynomial and of its derivatives up to order inclusive at the point coincide with the values of the function and of its corresponding derivatives at the same point:

The Taylor polynomial is the best polynomial approximation of the function as , in the sense that

(*)

and if some polynomial of degree not exceeding has the property that

where , then it coincides with the Taylor polynomial . In other words, the polynomial having the property (*) is unique.

If at least one of the derivatives , , is not equal to 0 at the point , then the Taylor polynomial is the principal part of the Taylor formula.

Comments

For references see Taylor formula.