Mathematik-Online-Kurs: Analysis einer Veränderlichen-Differentiation-Taylor-Entwicklung-Spezielle Taylor-Reihen

$\displaystyle (1+x)^\alpha$	$\displaystyle = \sum_{k=0}^\infty \binom{\alpha}{k}x^k = 1 + \alpha x + \frac{\alpha(\alpha-1)}{2!}x^2 + \frac{\alpha(\alpha-1)(\alpha-2)}{3!}x^3+\cdots$	$\displaystyle \vert x\vert<1$
$\displaystyle e^x$	$\displaystyle = \sum_{k=0}^\infty \frac{x^k}{k!} = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \cdots$	$\displaystyle x \in\mathbb{R}$
$\displaystyle \ln(1+x)$	$\displaystyle = \sum_{k=1}^\infty (-1)^{k+1}\frac{x^k}{k} = x -\frac{x^2}{2} + \frac{x^3}{3} - \frac{x^4}{4}\pm\cdots \hfill$	$\displaystyle -1<x\leq 1$
$\displaystyle \sin x$	$\displaystyle = \sum_{k=0}^\infty (-1)^k\frac{x^{2k+1}}{(2k+1)!} = x-\frac{x^3}{3!}+\frac{x^5}{5!}-\frac{x^7}{7!}\pm\cdots$	$\displaystyle x \in\mathbb{R}$
$\displaystyle \cos x$	$\displaystyle = \sum_{k=0}^\infty (-1)^k\frac{x^{2k}}{(2k)!} = 1-\frac{x^2}{2!}+\frac{x^4}{4!}-\frac{x^6}{6!}\pm\cdots$	$\displaystyle x \in\mathbb{R}$
$\displaystyle \tan x$	$\displaystyle = \sum_{k=1}^\infty (-1)^{k-1}\frac{2^{2k}(2^{2k}-1)}{(2k)!}B_{2k}x^{2k-1} = x + \frac{x^3}{3} + \frac{2x^5}{15} + \frac{17x^7}{315}+\cdots$	$\displaystyle \vert x\vert<\frac{\pi}{2}$
$\displaystyle \arcsin x$	$\displaystyle = \sum_{k=0}^\infty \frac{1\cdot3\cdot5\cdots(2k-1)}{2\cdot4\cdot... ...^{2k+1}}{2k+1} = x + \frac{x^3}{6} + \frac{3x^5}{40} + \frac{5x^7}{112} +\cdots$	$\displaystyle \vert x\vert<1$
$\displaystyle \arccos x$	$\displaystyle = \frac{\pi}{2} - \sum_{k=0}^\infty \frac{1\cdot3\cdot5\cdots(2k-... ...- \left( x + \frac{x^3}{6} + \frac{3x^5}{40} + \frac{5x^7}{112} +\cdots \right)$	$\displaystyle \vert x\vert<1$
$\displaystyle \arctan x$	$\displaystyle = \sum_{k=0}^\infty (-1)^k\frac{x^{2k+1}}{2k+1} = x - \frac{x^3}{3} + \frac{x^5}{5} - \frac{x^7}{7} \pm \cdots$	$\displaystyle \vert x\vert<1$

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	Mathematik-Online-Kurs: Analysis einer Veränderlichen - Differentiation - Taylor-Entwicklung
	Spezielle Taylor-Reihen