Taylorreihe und Reihen

Definition

$$f(x) = f(a)+\frac{x-a}{1!}f'(a)+\frac{(x-a)^2}{2!}f''(a)+\ldots+\frac{(x-a)^n}{n!}f^{(n)}(a)+\ldots$$

andere Schreibweise

$$f(x+\Delta x) = f(x)+\frac{\Delta x}{1!}f'(x)+\frac{(\Delta x)^2}{2!}f''(x)+\ldots+\frac{(\Delta x)^n}{n!}f^{(n)}(x)+\ldots$$

Beispiel

$$f(x+\Delta x) = \sin(x + \Delta x)$$

$$ = \sin(x)+\frac{\Delta x}{1!}\cos(x)+\frac{(\Delta x)^2}{2!}f''(x)+\ldots$$

$$ + (-1)^n \frac{(\Delta x)^{2n}}{(2n)!}\sin(x)+\ldots+ (-1)^n \frac{(\Delta x)^{2n+1}}{(2n+1)!}\cos(x)+\ldots$$

Spezialfall: $x=0$

$$\sin(\Delta x) = \Delta x - \frac{1}{3!} (\Delta x)^5 + \frac{1}{3!} (\Delta x)^5 +\ldots+(-1)^n \frac{(\Delta x)^{2n+1}}{(2n+1)!}+\ldots$$