[THEOREMS] Taylor's theorem

Taylor's theorem with remainder

Suppose $f$ has $n+1$ continuous derivatives on an interval containing $a$ and $x$ . If

P_n(x)=\sum_{k=0}^{n}\frac{f^{(k)}(a)}{k!}(x-a)^k,

then

f(x)=P_n(x)+R_{n+1}(x),

where the integral remainder is

R_{n+1}(x) =\frac1{n!}\int_a^x f^{(n+1)}(t)(x-t)^n\,dt.

For $x\ne a$ , there is also a point $c$ between $a$ and $x$ for which the Lagrange remainder is

R_{n+1}(x) =\frac{f^{(n+1)}(c)}{(n+1)!}(x-a)^{n+1}.

The integral form accumulates the first omitted derivative. The Lagrange form replaces the integral by one unknown derivative value. If

|f^{(n+1)}(t)|\le M

throughout the interval between $a$ and $x$ , then

|f(x)-P_n(x)| =|R_{n+1}(x)| \le \frac{M|x-a|^{n+1}}{(n+1)!}.

The bound depends on the distance $|x-a|$ from the centre and the size of the first omitted derivative. The factorial often dominates the numerator for large $n$ , but this must be checked for the interval and degree in use.

[1]:Taylor polynomial
[2]:When a Taylor series represents its function
[3]:Taylor series
[4]:Proof
[5]:The Mean Value Theorem as the first case
3 more

Classify a stationary point with the first nonzero derivative

Suppose $k$ is an integer with $k\ge2$ and $f$ has $k+1$ continuous derivatives on a neighbourhood of $a$ . If

f^{(j)}(a)=0\quad(1\le j<k), \qquad f^{(k)}(a)\ne0.

then Taylor's theorem gives

f(x)-f(a) =\frac{f^{(k)}(a)}{k!}(x-a)^k+R_{k+1}(x), \qquad R_{k+1}(x)=O\bigl(|x-a|^{k+1}\bigr),\qquad R_{k+1}(x)=o\bigl(|x-a|^k\bigr).

Thus, locally:

if $k$ is even and $f^{(k)}(a)>0$ , $a$ is a local minimum;
if $k$ is even and $f^{(k)}(a)<0$ , $a$ is a local maximum;
if $k$ is odd, the leading term changes sign across $a$ , giving a horizontal point of inflection.

Here $O(|x-a|^{k+1})$ means an error bounded by a constant times $|x-a|^{k+1}$ near $a$ and $o(|x-a|^k)$ means an error whose ratio to $|x-a|^k$ tends to zero.

The test requires $k\ge2$ , a genuinely first nonzero derivative and enough continuity to ensure that the remainder is $o(|x-a|^k)$ . If all derivatives examined vanish, those terms give no classification.

Theorems from Taylor's theorem

Taylor's theorem with remainder

Classify a stationary point with the first nonzero derivative

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