Wichtige Reihen

Lesedauer: 4 min | Vorlesen | Autor: Dr. Volkmar Naumburger

\( \sin (x) = 1 \cdot x - \frac{1}{6} \cdot {x^3} + \,\frac{1}{ {120} } \cdot {x^5}.... = \sum\limits_{n = 0}^\infty { { {\left( { - 1} \right)}^n}\frac{1}{ {(2n + 1)!} } \cdot {x^{2n + 1} } } ; \) r < ∞ Gl. 198

\( \cos (x) = 1 - \frac{1}{ {2!} } \cdot {x^2} + \,\frac{1}{ {4!} } \cdot {x^4} - \frac{1}{ {6!} } \cdot {x^6}.... = \sum\limits_{n = 0}^\infty { { {\left( { - 1} \right)}^n}\frac{1}{ {(2n)!} } \cdot {x^{2n} } } ; \) r < ∞ Gl. 199

\( {e^x} = 1 + \frac{1}{ {1!} } \cdot x + \,\frac{1}{ {2!} } \cdot {x^2} + \frac{1}{ {3!} } \cdot {x^3}.... = \sum\limits_{n = 0}^\infty {\frac{1}{ {n!} } \cdot {x^n} } ; \) r < ∞ Gl. 200

\( \arcsin (x) = x + \frac{1}{2}\left( {\frac{1}{3} \cdot {x^3} + \,\frac{ {1 \cdot 3} }{ {4 \cdot 5} } \cdot {x^5}.... + \frac{ {1 \cdot 3 \cdot ... \cdot (2n - 3)} }{ {4 \cdot ... \cdot (2n - 2) \cdot (2n - 1)} } \cdot {x^{2n - 1} } } \right); \) r < 1 Gl. 201

\( \arccos (x) = \frac{\pi }{2} - x - \frac{1}{2}\left( {\frac{1}{3} \cdot {x^3} + \,\frac{ {1 \cdot 3} }{ {4 \cdot 5} } \cdot {x^5}.... + \frac{ {1 \cdot 3 \cdot ... \cdot (2n - 3)} }{ {4 \cdot ... \cdot (2n - 2) \cdot (2n - 1)} } \cdot {x^{2n - 1} } } \right); \) r < 1 Gl. 202

\( \arctan (x) = x - \frac{1}{3} \cdot {x^3} + \,\frac{1}{5} \cdot {x^5} - \frac{1}{7} \cdot {x^7}.... + \frac{ { { {\left( { - 1} \right)}^n} } }{ {(2n + 1)} } \cdot {x^{2n + 1} }; \) r < 1 Gl. 203

\( \ln (1 + x) = x - \,\frac{1}{2} \cdot {x^2} + \frac{1}{3} \cdot {x^3} - \frac{1}{4} \cdot {x^4}.... = \sum\limits_{n = 0}^\infty {\frac{1}{n} \cdot {x^n} } ; \) r < 1 Gl. 204

Binomische Reihe

\( {(1 + x)^m} = 1 + \,\left( {\begin{array}{*{20}{c} }m\\1\end{array} } \right) \cdot {x^1} + \,\left( {\begin{array}{*{20}{c} }m\\2\end{array} } \right) \cdot {x^2} + \left( {\begin{array}{*{20}{c} }m\\3\end{array} } \right) \cdot {x^3}.... = \sum\limits_{n = 0}^m {\left( {\begin{array}{*{20}{c} }m\\n\end{array} } \right) \cdot {x^n} } ; \) r < 1 Gl. 205

Achtung: die Binomische Reihe gilt auch für alle reellen m.

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