Taylor series and error estimation

3.6 Taylor series and error estimation

If $f(x)$ is a ”normal” function than \begin{eqnarray*} f(x)=f(x_0)&+&\frac{1}{1!}(x-x_0)f'(x_0)+\frac{1}{2!}(x-x_0)f''(x_0)+\ldots+\frac{(x-x_0)^{n-1}}{(n-1)!}f^{n-1}(x_0)\\ &+&\frac{(x-x_0)^n}{n!}f^n(x_0)+\frac{(x-x_0)^{n+1}}{(n+1)!}f^{n+1}(x_0+\epsilon\Delta x)\end{eqnarray*}

i.e. the error term

\[\Delta f=\frac{(x-x_0)^{n+1}}{(n+1)!}f^{n+1}(x_0+\epsilon\Delta x)\quad\begin{array}{ccl}\Delta x&=&x-x_0\\0&\lt&\varepsilon\lt1\end{array}\]

is the difference between the exact function and the approximation.
Example: \begin{eqnarray*}f(x)&=&\sin x;\qquad x_0=0\\ \mbox{Taylor:}\;f(x)&=&x-\frac{x^3}{3!}+\frac{x^5}{5!}f^5(\epsilon x)\qquad 0\lt\epsilon\lt1\\ f^5(x)&=&\cos x\Rightarrow\,\left|f(x)-\left(x-\frac{x^3}{3!}\right)\right|=|\Delta f|=\frac{x^5}{5!}|\cos(\epsilon x)|\le\frac{x^5}{5!}\rightarrow\mbox{may diverge for large $x$}\\ \mbox{but for: } x&=&10^\circ=\underbrace{\frac{10\pi}{180}}_{0.17}\rightarrow\left|f(x)-(x-\frac{x^3}{3!})\right|\le10^{-6} \end{eqnarray*}

Definition 32

\[\sum_{k=0}^n\frac{f^k(x_0)}{k!}(x-x_0)^k\]

is called a Taylor approximation or series of the order $n$ to the function $f(x)$ at the point $x_0$.

Example: Approximation of third order to $f(x)=e^x$ at $x_0=0$:

\[e^x\approx1+x+\frac{1}{2}x^2+\frac{1}{6}x^3+\mbox{O}(x^4)\]

notation ”zero of $x^4$” means approximately up to $x^3$ exact
Sometimes problems: (rare cases)

$f(x)^=e^{-\frac{1}{x^2}}\;$ if $x\neq0$ and $f(x)=0$ if $x=0$. $\rightarrow$ well defined function.

Taylor-Series around $x_0=0$ \begin{eqnarray*} f'(x)&=&\frac{2}{x^3}e^{-\frac{1}{x^2}}x+0;\;\;x=0:\;\;\frac{f(x)-f(0)}{x-0}=\frac{e^{-\frac{1}{x^2}}}{x}\\ \frac{1}{x}&=&m\rightarrow\;\frac{f(x)-f(0)}{x-0}=\frac{e^{-m^2}}{\frac{1}{m}}=me^{-m^2}\rightarrow0\quad\mbox{for $m\to\infty$}\\ &\Rightarrow&f'(0)=0\quad\mbox{and also }f^n(0)=0\quad\mbox{for all $n$!!}\\ &\Rightarrow&\mbox{Thus Taylor-series of $f(x)$ is $\equiv0$}\Rightarrow\mbox{unsuccessful approximation!!} \end{eqnarray*}

3.6.1 Example: Transformation of a non linear problem into a linear problem
3.6.2 Example: Taylor series of arctan function and pi-calculation