`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).
`