Formal Statement of Taylor's Theorem
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:
![\begin{eqnarray*}
f(x) = f(x_0) &+& \frac{1}{1}f^\prime(x_0)(x-x_0) \\ [10pt]
&...
...&+& \frac{1}{n!}f^{(n+1)}(x_0)(x-x_0)^n\\ [10pt]
&+& R_{n+1}(x)
\end{eqnarray*}](http://www.dsprelated.com/josimages_new/mdft/img1874.png)
where
is called the remainder term. Then Taylor's
theorem [63, pp. 95-96] provides that there exists some
between
and
such that






When , the Taylor series reduces to what is called a Maclaurin
series [56, p. 96].
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Weierstrass Approximation Theorem
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Taylor Series with Remainder