Any sufficiently smooth function can be approximated the order Taylor polynomial for
For example, if
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).