Let
be continuous on a real interval
containing
(and
),
and let
exist at
and
be continuous for
all
. Then we have the following Taylor series expansion:
where
is called the remainder term. Then Taylor's
theorem [66, pp. 95-96] provides that there exists some
between
and
such that
In particular, if
which is normally small when
When
, the Taylor series reduces to what is called a Maclaurin
series [59, p. 96].