The relevant information is listed in the following table

Using the formula above we obtain as the second degree Taylor polynomial:

No reason to only compute second degree Taylor polynomials!
If we want to find for example the fourth degree Taylor polynomial for a function *f*(*x*) with a given center , we will insist that the polynomial and *f*(*x*) have the same value and the same first four derivatives at .

A calculation similar to the previous one will yield the formula:

(2) The factorial sign comes in handy: Recall that . For instance Mathematicians usually say that 0!=1.

(3) Recall that one writes for the *n*th derivative of the function*f*. just means *f*(*x*).

With these notations, we can then write the *n*th term of a Taylor polynomial as

Thus we obtain the general formula for the *n*th Taylor polynomial of a function *f*(*x*) with center :

An alternative way of writing this--for the not easily *Mathematese*-intimidated-- is provided by the summation notation:

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Sun Jul 7 22:08:09 MDT 1996

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