Taylor polynomials

Any sufficiently smooth function can be approximated the order Taylor polynomial for

The term is called the remainder, and measures the error of the Taylor approximation.

For example, if

then can be approximated along the interval by the following Taylor polynomial:

The quantity in the definition of is a number between and .

When we replace by we make an error of . An upper bound for this error is obtained as follows

We know that the factorial grows much faster than the exponential; in the above formula, when increases, the denominator grows faster than the numerator. Therefore it is clear that when (the order of the Taylor polynomial) increases, decreases (that is, the higher the order the better we approximate the function).