Let f be a function with f', f'',... and defined at a. the nth-order Taylor polynomial for f with its center at a, denoted has the property that it matches f in value, slope, and all derivatives up to the nth derivative at a; that is,
, ,...., and
The nth-order Taylor polynomial centered at a is
More completely, , where the coefficients are
, for k=0, 1, 2, ..... , n

Remainder of the Taylor Polynomial

Let be the Taylor polynomial of order n for f. The remainder in using to approximate f at the point x

Taylor Remainder Theorem

Let f have continuous derivatives up to on an open interval I containing a. For all x in I.
Where is the nth order Taylor polynomial for f centered at a and the remainder is
for some point c between x and a.

Estimate of the Remainder

Let n be a fixed positive integer. Suppose there exists a number M such that , for all c between a and x inclusive. The remainder in the nth-order Taylor polynomial for f centered at a satisfies