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Taylor polynomial

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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.

How to Cite This Entry:
Taylor polynomial. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Taylor_polynomial&oldid=29524
This article was adapted from an original article by L.D. Kudryavtsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article