Taylor Series

Taylor Series approximate functions of harshly accurately determinable nature and values. It gives a convenient polynomial that approximates the series and which’s error bound can also be determined if certain properties are known. The Taylor Series about a particular point $a$ is

$$f(a)+\frac{f^{\prime }(a)}{1!}(x-a)+\frac{f^{\prime \prime }(a)}{2!}(x-a)+...=\sum \_{n=0}^{\infty }\frac{f^{(n)}(a)}{n!}(x-a{)}^n$$

Error Bound Approximation using Remainder (n + 1)

Taylor-Polynom und Remainder (um a)

$$T\_n(x);=;\sum \_{k=0}^n\frac{f^{(k)}(a)}{k!},(x-a{)}^k$$

$\text{Def.: }R\_n(x);=;f(x)-T\_n(x)$

Lagrange-Form des Remainders

$$R\_n(x);=;\frac{f^{(n+1)}(\xi )}{(n+1)!},(x-a{)}^{,n+1}\text{fu¨r ein }\xi \text{ zwischen }a\text{ und }x.$$

Integral-Form des Remainders

$$R\_n(x);=;\frac{1}{n!}\int \_a^x{f}^{(n+1)}(t),(x-t{)}^{,n},dt$$

Taylor-Ungleichung (Fehlerabschätzung)

$$\left|R\_n(x)\right|;\le ;\frac{M}{(n+1)!},|x-a{|}^{,n+1},M;=;\max \_t\in I\left|f^{(n+1)}(t)\right|,\text{ wobei }I\text{ das Segment zwischen }a\text{ und }x\text{ ist.}$$