We have a function and we want to approximate it using anth-order polynomial:
where , which is obviously the approximation error,
is called the ''remainder term.''
We may assume and are real, but the following
derivation generalizes unchanged to the complex case.
Our problem is to find fixed constants so as to obtain
the best approximation possible. Let's proceed optimistically as though
the approximation will be perfect, and assume for all (), given the right values of . Then at we
must have
That's one constant down and to go! Now let's look at the first
derivative of with respect to , again assuming that
:
Evaluating this at gives
In the same way, we find
where denotes the th derivative of with respect to, evaluated at . Solving the above relations for the desired
constants yields
This is the th-order Taylor series expansion of about the point. Its derivation was quite simple. The hard part is showing that
the approximation error (remainder term ) is small over a wide
interval of values. Another ''math job'' is to determine the
conditions under which the approximation error approaches zero for all as the order goes to infinity. The main point to note here is that the
form of the Taylor series expansion itself is simple to derive.
and we want to approximate it using an
th-order polynomial:
, which is obviously the approximation error, is called the ''remainder term.'' We may assume
and
so as to obtain the best approximation possible. Let's proceed optimistically as though the approximation will be perfect, and assume
for all
), given the right values of
. Then at
we must have
to go! Now let's look at the first derivative of
denotes the
(as it always is), we have derived the following polynomial approximation:
