6. Taylor's Inequality

Approximation Error

Definition

Given a function $f$ and its Taylor polynomial $T_n$ at the point $a$, the approximation error in $x$ is given by

$R_n(x)=|f(x)-T_n(x)|$

In other words, $R_n(x)$ is the difference between the real function value and the value of the Taylor polynomial at the point $x$.

Taylor’s Inequality

Theorem

Let $f$ be a function and $T_n$ its Taylor polynomial at the point $a$.

Let $D$ be an interval that contains the point $a$ and $M$ an upper bound for the function $|f^{(n+1)}(x)|$ on $D$, in other words:

$|f^{(n+1)}(x)|\le M$ holds for all $x$ in $D$.

Then, for all $x\in D$:

$|f(x)-T_n(x)|\le \frac{M}{(n+1)!}|x-a{|}^{n+1}$

The section $|f(x)-T_n(x)|$ is also $R_n(x)$