Taylor's Theorem

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Theorem

Every infinitely differentiable function can be approximated by a series of polynomials.


One Variable

Let $f$ be a real function which is:

of differentiability class $C^n$ on the closed interval $\closedint a x$

and:

at least $n + 1$ times differentiable on the open interval $\openint a x$.

Then:

\(\displaystyle \map f x\) \(=\) \(\displaystyle \frac 1 {0!} \map f a\)
\(\displaystyle \) \(\) \(\, \displaystyle + \, \) \(\displaystyle \frac 1 {1!} \paren {x - a} \map {f'} a\)
\(\displaystyle \) \(\) \(\, \displaystyle + \, \) \(\displaystyle \frac 1 {2!} \paren {x - a}^2 \map {f''} a\)
\(\displaystyle \) \(\) \(\, \displaystyle + \, \) \(\displaystyle \cdots\)
\(\displaystyle \) \(\) \(\, \displaystyle + \, \) \(\displaystyle \frac 1 {n!} \paren {x - a}^n \map {f^{\paren n} } a\)
\(\displaystyle \) \(\) \(\, \displaystyle + \, \) \(\displaystyle R_n\)

where $R_n$ (sometimes denoted $E_n$) is known as the error term or remainder, and can be presented in one of $2$ forms:

Lagrange Form
$R_n = \dfrac {\map {f^{\paren {n + 1} } } \xi} {\paren {n + 1}!} \paren {x - a}^{n + 1}$

for some $\xi \in \openint a x$.

Cauchy Form
$R_n = \dfrac {\map {f^{\paren {n + 1} } } \xi \paren {x - \xi}^n} {n!} \paren {x - a}$

for some $\xi \in \openint a x$.


One Variable with Two Functions

Let $f$ and $g$ be real functions satisfying following conditions:

$(1): \quad f$ is $n + 1$ times differentiable on the open interval $\openint a x$
$(2): \quad f$ is of differentiability class $C^n$ on the closed interval $\closedint a x$
$(3): \quad g$ is $k + 1$ times differentiable on the open interval $\openint a x$
$(4): \quad g$ is of differentiability class $C^k$ on the closed interval $\closedint a x$
$(5): \quad \map {g^{\paren {k + 1}}} t \ne 0$ for any $t \in \openint a x$


Then the following equation holds for some real number $\xi \in \openint a x$:

$\dfrac {\map {f^{\paren {n + 1} } } \xi /n!} {\map {g^{\paren {k + 1} } } \xi /k!} \paren {x - \xi}^{n - k} = \dfrac {\map f x - \map f a - \map {f'} a \paren {x - a} - \dfrac {\map {f''} a} {2!} \paren {x - a}^2 - \dotsb - \dfrac {\map {f^{\paren n} } a} {n!} \paren {x - a}^n} {\map g x - \map g a - \map {g'} a \paren {x - a} - \dfrac {\map {g''} a} {2!} \paren {x - a}^2 - \dotsb - \dfrac {\map {g^{\paren k} } a} {k!} \paren {x - a}^k}$

or equivalently:

\(\displaystyle \map f x\) \(=\) \(\displaystyle \map f a + \map {f'} a \paren {x - a} + \dfrac {\map {f''} a} {2!} \paren {x - a}^2 + \dotsb + \dfrac {\map {f^{\paren n} } a} {n!} \paren {x - a}^n + R_n\)
\(\displaystyle R_n\) \(=\) \(\displaystyle \dfrac {\map {f^{\paren {n + 1} } } \xi / n!} {\map {g^{\paren {k + 1} } } \xi / k!} \paren {x - \xi}^{n - k} \paren {\map g x - \map g a - \map {g'} a \paren {x - a} - \dfrac {\map {g''} a} {2!} \paren {x - a}^2 - \dotsb - \dfrac {\map {g^{\paren k} } a} {k!} \paren {x - a}^k}\)


Source of Name

This entry was named for Brook Taylor.


Historical Note

Taylor's Theorem, as applied to an analytic function, was estabished by Carl Friedrich Gauss in $1831$, but he never got round to publishing this work.