View source for Taylor polynomial

''of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092310/t0923101.png" />, for a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092310/t0923102.png" /> that is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092310/t0923103.png" /> times differentiable at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092310/t0923104.png" />''

The polynomial

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092310/t0923105.png" /></td> </tr></table>

The values of the Taylor polynomial and of its derivatives up to order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092310/t0923106.png" /> inclusive at the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092310/t0923107.png" /> coincide with the values of the function and of its corresponding derivatives at the same point:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092310/t0923108.png" /></td> </tr></table>

The Taylor polynomial is the best polynomial approximation of the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092310/t0923109.png" /> as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092310/t09231010.png" />, in the sense that

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092310/t09231011.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>

and if some polynomial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092310/t09231012.png" /> of degree not exceeding <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092310/t09231013.png" /> has the property that

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092310/t09231014.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092310/t09231015.png" />, then it coincides with the Taylor polynomial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092310/t09231016.png" />. In other words, the polynomial having the property (*) is unique.

If at least one of the derivatives <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092310/t09231017.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092310/t09231018.png" />, is not equal to 0 at the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092310/t09231019.png" />, then the Taylor polynomial is the principal part of the [[Taylor formula|Taylor formula]].



====Comments====
For references see [[Taylor formula|Taylor formula]].