Benutzer:Pendethan/Meine Entwürfe/Formeln

siehe Hilfe:TeX

$Saturn\quad \alpha \ {\text{Saturn}}\qquad \mathbf {Saturn} \ {\mathtt {Saturn}}\ {\boldsymbol {Saturn}}\ {\mathfrak {Saturn}}$

$\zeta =2\,\cdot \,\arcsin \left(\,{\frac {1}{2}}\cdot {\sqrt {{\Big (}\sin(\phi _{A})-\sin(\phi _{B}){\Big )}^{2}+{\Big (}\cos(\phi _{A})\cdot \cos(\lambda _{A})+\cos(\phi _{B})\cdot \cos(\lambda _{B}){\Big )}^{2}+{\Big (}\cos(\phi _{A})\cdot \sin(\lambda _{A})-\cos(\phi _{B})\cdot \sin(\lambda _{B}){\Big )}^{2}}}\ \right)$

$\zeta =2\,\cdot \,\arcsin \left(\,{\frac {1}{2}}\cdot {\sqrt {{\Big (}\sin(\phi _{A})-\sin(\phi _{B}){\Big )}^{2}+{\Big (}\cos(\phi _{A})\cdot \cos(\lambda _{A})+\cos(\phi _{B})\cdot \cos(\lambda _{B}){\Big )}^{2}+{\Big (}\cos(\phi _{A})\cdot \sin(\lambda _{A})-\cos(\phi _{B})\cdot \sin(\lambda _{B}){\Big )}^{2}}}\ \right)$

${\begin{alignedat}{2}&with:\ &&x=\cos \left(\delta \right)\cdot \cos \left(\alpha \right)\\&&&y=\cos \left(\delta \right)\cdot \sin \left(\alpha \right)\\&&&z=\sin \left(\delta \right)\\{\begin{aligned}&\alpha '=\arccos \left(\cos \left(\delta \right)\cdot \cos \left(\alpha \right)\right)\\&\alpha '=\arctan \left(y'/x\right)\end{aligned}}&yes&&y'=z\cdot \cos \left(23.5^{\circ }\right)-y\cdot \sin \left(23.5^{\circ }\right)\end{alignedat}}$

${\begin{aligned}&\alpha '=\arccos \left(\cos \left(\delta \right)\cdot \cos \left(\alpha \right)\right)\\&\alpha '=\arctan \left(y'/x\right)\end{aligned}}$

${\begin{alignedat}{2}&with:\ &&x=\cos \left(\delta \right)\cdot \cos \left(\alpha \right)\\&&&y=\cos \left(\delta \right)\cdot \sin \left(\alpha \right)\\&&&z=\sin \left(\delta \right)\\&&&y'=z\cdot \cos \left(23.5^{\circ }\right)-y\cdot \sin \left(23.5^{\circ }\right)\end{alignedat}}$

Exponentiation is a mathematical operation, written as $b^{n}$, scriptstyle: $\scriptstyle b^{n}$, displaystyle:

$\displaystyle b^{n}$, textstyle: $\textstyle b^{n}$, involving two numbers, the base $b$ and the exponent $n$. When $n$ is a positive integer, exponentiation corresponds to repeated multiplication of the base: that is, $b^{n}$ is the product of multiplying $n$ bases: $b^{n}=\underbrace {b\times \cdots \times b} _{n}$, scriptstyle: $\scriptstyle b^{n}=\underbrace {b\times \cdots \times b} _{n}$, displaystyle:
$\displaystyle b^{n}=\underbrace {b\times \cdots \times b} _{n}$, textstyle: $\textstyle b^{n}=\underbrace {b\times \cdots \times b} _{n}$

$\displaystyle b^{n}=\underbrace {b\times \cdots \times b} _{n}$

${\frac {2+{\frac {1/2+{\frac {5}{3}}}{5^{3}}}}{\sqrt {\frac {2}{3}}}}$

The exponent is usually shown as a superscript to the right of the base. In that case, $\scriptstyle b^{n}$ is called “b raised to the n-th power”, b raised to the power of n, or the n-th power of b.

When $n$ is a positive integer and $\textstyle n$ is not zero, $\textstyle b^{-n}$ is naturally defined as ${\textstyle {\frac {1}{b^{n}}}}$ or $\textstyle 1/b^{n}$, or $\textstyle {\frac {1}{b^{n}}}$ preserving the property $\scriptstyle b^{n}\times b^{m}=b^{n+m}$. With exponent $\scriptscriptstyle -1$ $\scriptstyle -1$, $\textstyle b^{-1}$ is equal to $\scriptstyle 1/b$, and is the reciprocal of $\textstyle b$. $\textstyle \int _{a}^{b}$

log_b x – Logarithmus zu einer beliebigen Basis b berechnen

${\begin{alignedat}{2}{\text{es sei}}\\y&=\log _{b}x&\qquad &{\big |}\ {\text{auf beiden Seiten}}\ b^{(\ )}\ {\text{und mit}}\ b^{\log _{b}x}=x\\b^{y}&=x&&{\big |}\ {\text{auf beiden Seiten}}\log _{a}{(\ )}\\\log _{a}b^{y}&=\log _{a}x&&{\big |}\ {\text{mit}}\log b^{y}=y\cdot \log a\\y\log _{a}b&=\log _{a}x&&{\big |}\ {\text{auf beiden Seiten}}\div \log _{a}b\\{\text{dann ist}}\\y=\log _{b}x&={\frac {\log _{a}x}{\log _{a}b}}\end{alignedat}}$

Division

Find the equation of the line that is tangent to the following curve at $x=1$:

$y=x^{3}-12x^{2}-42.$

Begin by dividing the polynomial by $(x-1)^{2}=x^{2}-2x+1$:

${\begin{array}{rclcl}x^{3}-12x^{2}{\color {Red}\,+\,21x}-42&\div &x^{2}-2x+1&=&x\\x^{3}-12x^{2}+21x-42\\{\underline {x^{3}-{\color {Yellow}0}2x^{2}+{\color {Red}1}x\,{\color {Red}-\,42}}}\\-10x^{2}-{\color {White}01}x-42\\{\underline {-10x^{2}+20x-10}}\\-21x-32\end{array}}$

Drehmatrix

Drehung um die $x$-Achse:

$R_{x}(\alpha )={\begin{pmatrix}1&0&0\\0&\cos \alpha &-\sin \alpha \\0&\sin \alpha &\cos \alpha \end{pmatrix}}$

Drehung um die $y$- und die $z$-Achse:

$R_{y}(\alpha )={\begin{pmatrix}\cos \alpha &0&\sin \alpha \\0&1&0\\-\sin \alpha &0&\cos \alpha \end{pmatrix}};\quad R_{z}(\alpha )={\begin{pmatrix}\cos \alpha &-\sin \alpha &0\\\sin \alpha &\cos \alpha &0\\0&0&1\end{pmatrix}}$

${\begin{aligned}&a^{(i)}=[{}_{i}^{k}A]\cdot a^{(k)}\\&a^{(k)}=[{}_{i}^{k}A]^{-1}\cdot a^{(i)}\end{aligned}}$

Inverse Drehmatrizen:

$(A\cdot B\cdot C)^{-1}=C^{-1}\cdot B^{-1}\cdot A^{-1}$

$Pose={\begin{pmatrix}r_{00}&r_{01}&r_{02}&t_{0}\\r_{10}&r_{11}&r_{12}&t_{1}\\r_{20}&r_{21}&r_{22}&t_{2}\\0&0&0&1\end{pmatrix}}$

Kombinierte, hintereinander ausgeführte (innere) Drehungen um die Achsen $x$, $y$ und $z$:

${\begin{pmatrix}r_{00}&r_{01}&r_{02}\\r_{10}&r_{11}&r_{12}\\r_{20}&r_{21}&r_{22}\end{pmatrix}}={\begin{pmatrix}\cos \theta _{y}\cdot \cos \theta _{z}&-\cos \theta _{y}\cdot \sin \theta _{z}&\sin \theta _{y}\\\sin \theta _{x}\cdot \sin \theta _{y}\cdot \cos \theta _{z}+\cos \theta _{x}\cdot \sin \theta _{z}&-\sin \theta _{x}\cdot \sin \theta _{y}\cdot \sin \theta _{z}+\cos \theta _{x}\cdot \cos \theta _{z}&-\sin \theta _{x}\cdot \cos \theta _{y}\\-\cos \theta _{x}\cdot \sin \theta _{y}\cdot \cos \theta _{z}+\sin \theta _{x}\cdot \sin \theta _{z}&\cos \theta _{x}\cdot \sin \theta _{y}\cdot \sin \theta _{z}+\sin \theta _{x}\cdot \cos \theta _{z}&\cos \theta _{x}\cdot \cos \theta _{y}\end{pmatrix}}$

$r_{02}=\sin \theta _{y}\quad \Longrightarrow \quad \theta _{y}={\text{asin}}(r_{02})$

${\text{(1) für}}\quad \cos \theta _{y}\neq 0\quad \Longrightarrow \quad \theta _{y}\neq 90^{\circ }\quad {\text{und}}\quad \theta _{y}\neq -90^{\circ }$

$\Longrightarrow \quad \sin \theta _{x}={\frac {-r_{12}}{\cos \theta _{y}}}\quad {\text{und}}\quad \cos \theta _{x}={\frac {r_{22}}{\cos \theta _{y}}};\quad \sin \theta _{z}={\frac {-r_{01}}{\cos \theta _{y}}}\quad {\text{und}}\quad \cos \theta _{z}={\frac {r_{00}}{\cos \theta _{y}}}$

$\Longrightarrow \quad \theta _{x}={\text{atan2}}(-r_{12},r_{22});\quad \theta _{z}={\text{atan2}}(-r_{01},r_{00})\quad {\text{mit}}\quad \theta ={\text{atan2}}(\sin \theta ,\cos \theta )$

${\text{(2) für}}\quad \theta _{y}=90^{\circ }\quad \Longrightarrow \quad \sin \theta _{y}=1;\quad \cos \theta _{y}=0\quad \Longrightarrow \quad r_{00}=0;\quad r_{01}=0;\quad r_{02}=1;\quad r_{12}=0;\quad r_{22}=0$

${\begin{alignedat}{3}&r_{10}=r_{21}&&=\sin \theta _{x}\cdot \cos \theta _{z}+\cos \theta _{x}\cdot \sin \theta _{z}\quad &&\Longrightarrow \quad r_{10}=r_{21}=\sin(\theta _{x}+\theta _{z})\\&r_{11}=-r_{20}&&=\cos \theta _{x}\cdot \cos \theta _{z}-\sin \theta _{x}\cdot \sin \theta _{z}\quad &&\Longrightarrow \quad r_{11}=-r_{20}=\cos(\theta _{x}+\theta _{z})\end{alignedat}}$

$\Longrightarrow \quad \theta _{x}+\theta _{z}={\text{atan2}}(r_{10},r_{11})={\text{atan2}}(r_{21},-r_{20})\quad {\text{mit}}\quad \theta ={\text{atan2}}(\sin \theta ,\cos \theta )\quad {\text{Lösung mit einem Freiheitsgrad,}}\ \theta _{x}\ {\text{oder}}\ \theta _{z}\ {\text{ist frei wählbar}}$

${\text{(3) für}}\quad \theta _{y}=-90^{\circ }\quad \Longrightarrow \quad \sin \theta _{y}=-1;\quad \cos \theta _{y}=0\quad \Longrightarrow \quad r_{00}=0;\quad r_{01}=0;\quad r_{02}=-1;\quad r_{12}=0;\quad r_{22}=0$

${\begin{alignedat}{5}-r_{10}&=r_{21}&&=\sin \theta _{x}\cdot \cos \theta _{z}-\cos \theta _{x}\cdot \sin \theta _{z}\quad &&\Longrightarrow \quad -r_{10}&&=r_{21}&&=\sin(\theta _{x}-\theta _{z})\\r_{11}&=r_{20}&&=\cos \theta _{x}\cdot \cos \theta _{z}+\sin \theta _{x}\cdot \sin \theta _{z}\quad &&\Longrightarrow \quad r_{11}&&=r_{20}&&=\cos(\theta _{x}-\theta _{z})\end{alignedat}}$

$\Longrightarrow \quad \theta _{x}-\theta _{z}={\text{atan2}}(-r_{10},r_{11})={\text{atan2}}(r_{21},r_{20})\quad {\text{mit}}\quad \theta ={\text{atan2}}(\sin \theta ,\cos \theta )\quad {\text{Lösung mit einem Freiheitsgrad,}}\ \theta _{x}\ {\text{oder}}\ \theta _{z}\ {\text{ist frei wählbar}}$

Kombinierte, hintereinander ausgeführte (innere) Drehungen um die Achsen $z$, $y$ und $x$:

${\begin{pmatrix}r_{00}&r_{01}&r_{02}\\r_{10}&r_{11}&r_{12}\\r_{20}&r_{21}&r_{22}\end{pmatrix}}={\begin{pmatrix}\cos \theta _{y}\cdot \cos \theta _{z}&\sin \theta _{x}\cdot \sin \theta _{y}\cdot \cos \theta _{z}-\cos \theta _{x}\cdot \sin \theta _{z}&\cos \theta _{x}\cdot \sin \theta _{y}\cdot \cos \theta _{z}+\sin \theta _{x}\cdot \sin \theta _{z}\\\cos \theta _{y}\cdot \sin \theta _{z}&\sin \theta _{x}\cdot \sin \theta _{y}\cdot \sin \theta _{z}+\cos \theta _{x}\cdot \cos \theta _{z}&\cos \theta _{x}\cdot \sin \theta _{y}\cdot \sin \theta _{z}-\sin \theta _{x}\cdot \cos \theta _{z}\\-\sin \theta _{y}&\sin \theta _{x}\cdot \cos \theta _{y}&\cos \theta _{x}\cdot \cos \theta _{y}\end{pmatrix}}$

$-r_{20}=\sin \theta _{y}\quad \Longrightarrow \quad \theta _{y}={\text{asin}}(-r_{20})$

${\text{(1) für}}\quad \cos \theta _{y}\neq 0\quad \Longrightarrow \quad \theta _{y}\neq 90^{\circ }\quad {\text{und}}\quad \theta _{y}\neq -90^{\circ }$

$\Longrightarrow \quad \sin \theta _{x}={\frac {r_{21}}{\cos \theta _{y}}}\quad {\text{und}}\quad \cos \theta _{x}={\frac {r_{22}}{\cos \theta _{y}}};\quad \sin \theta _{z}={\frac {r_{10}}{\cos \theta _{y}}}\quad {\text{und}}\quad \cos \theta _{z}={\frac {r_{00}}{\cos \theta _{y}}}$

$\Longrightarrow \quad \theta _{x}={\text{atan2}}(r_{21},r_{22});\quad \theta _{z}={\text{atan2}}(r_{10},r_{00})\quad {\text{mit}}\quad \theta ={\text{atan2}}(\sin \theta ,\cos \theta )$

${\text{(2) für}}\quad \theta _{y}=90^{\circ }\quad \Longrightarrow \quad \sin \theta _{y}=1;\quad \cos \theta _{y}=0\quad \Longrightarrow \quad r_{00}=0;\quad r_{10}=0;\quad r_{20}=1;\quad r_{21}=0;\quad r_{22}=0$

${\begin{alignedat}{3}&r_{01}=-r_{12}&&=\sin \theta _{x}\cdot \cos \theta _{z}-\cos \theta _{x}\cdot \sin \theta _{z}\quad &&\Longrightarrow \quad r_{01}=-r_{12}=\sin(\theta _{x}-\theta _{z})\\&r_{11}=r_{02}&&=\cos \theta _{x}\cdot \cos \theta _{z}+\sin \theta _{x}\cdot \sin \theta _{z}\quad &&\Longrightarrow \quad r_{11}=r_{02}=\cos(\theta _{x}-\theta _{z})\end{alignedat}}$

$\Longrightarrow \quad \theta _{x}-\theta _{z}={\text{atan2}}(r_{01},r_{11})={\text{atan2}}(-r_{12},r_{02})\quad {\text{mit}}\quad \theta ={\text{atan2}}(\sin \theta ,\cos \theta )\quad {\text{Lösung mit einem Freiheitsgrad,}}\ \theta _{x}\ {\text{oder}}\ \theta _{z}\ {\text{ist frei wählbar}}$

${\text{(3) für}}\quad \theta _{y}=-90^{\circ }\quad \Longrightarrow \quad \sin \theta _{y}=-1;\quad \cos \theta _{y}=0\quad \Longrightarrow \quad r_{00}=0;\quad r_{10}=0;\quad r_{20}=-1;\quad r_{21}=0;\quad r_{22}=0$

${\begin{alignedat}{5}-r_{01}&=-r_{12}&&=\sin \theta _{x}\cdot \cos \theta _{z}+\cos \theta _{x}\cdot \sin \theta _{z}\quad &&\Longrightarrow \quad -r_{01}&&=-r_{12}&&=\sin(\theta _{x}+\theta _{z})\\r_{11}&=-r_{02}&&=\cos \theta _{x}\cdot \cos \theta _{z}-\sin \theta _{x}\cdot \sin \theta _{z}\quad &&\Longrightarrow \quad r_{11}&&=-r_{02}&&=\cos(\theta _{x}+\theta _{z})\end{alignedat}}$

$\Longrightarrow \quad \theta _{x}+\theta _{z}={\text{atan2}}(-r_{01},r_{11})={\text{atan2}}(-r_{12},-r_{02})\quad {\text{mit}}\quad \theta ={\text{atan2}}(\sin \theta ,\cos \theta )\quad {\text{Lösung mit einem Freiheitsgrad,}}\ \theta _{x}\ {\text{oder}}\ \theta _{z}\ {\text{ist frei wählbar}}$

Kombinierte, hintereinander ausgeführte (innere) Drehungen um die Achsen $z$, $y$ und $z$:

${\begin{pmatrix}r_{00}&r_{01}&r_{02}\\r_{10}&r_{11}&r_{12}\\r_{20}&r_{21}&r_{22}\end{pmatrix}}={\begin{pmatrix}\cos \theta _{z_{1}}\cdot \cos \theta _{y}\cdot \cos \theta _{z_{2}}-\sin \theta _{z_{1}}\cdot \sin \theta _{z_{2}}&-\cos \theta _{z_{1}}\cdot \cos \theta _{y}\cdot \sin \theta _{z_{2}}-\sin \theta _{z_{1}}\cdot \cos \theta _{z_{2}}&\cos \theta _{z_{1}}\cdot \sin \theta _{y}\\\sin \theta _{z_{1}}\cdot \cos \theta _{y}\cdot \cos \theta _{z_{2}}+\cos \theta _{z_{1}}\cdot \sin \theta _{z_{2}}&-\sin \theta _{z_{1}}\cdot \cos \theta _{y}\cdot \sin \theta _{z_{2}}+\cos \theta _{z_{1}}\cdot \cos \theta _{z_{2}}&\sin \theta _{z_{1}}\cdot \sin \theta _{y}\\-\sin \theta _{y}\cdot \cos \theta _{z_{2}}&\sin \theta _{y}\cdot \sin \theta _{z_{2}}&\cos \theta _{y}\end{pmatrix}}$

$r_{22}=\cos \theta _{y}\quad \Longrightarrow \quad \theta _{y}={\text{acos}}(r_{22})$

${\text{(1) für}}\quad \sin \theta _{y}\neq 0\quad \Longrightarrow \quad \theta _{y}\neq 0^{\circ }\quad {\text{und}}\quad \theta _{y}\neq 180^{\circ }$

$\Longrightarrow \quad \sin \theta _{z_{1}}={\frac {r_{12}}{\sin \theta _{y}}}\quad {\text{und}}\quad \cos \theta _{z_{1}}={\frac {r_{02}}{\sin \theta _{y}}};\quad \sin \theta _{z_{2}}={\frac {r_{21}}{\sin \theta _{y}}}\quad {\text{und}}\quad \cos \theta _{z_{2}}={\frac {-r_{20}}{\sin \theta _{y}}}$

$\Longrightarrow \quad \theta _{z_{1}}={\text{atan2}}(r_{12},r_{02});\quad \theta _{z_{2}}={\text{atan2}}(r_{21},-r_{20})\quad {\text{mit}}\quad \theta ={\text{atan2}}(\sin \theta ,\cos \theta )$

${\text{(2) für}}\quad \theta _{y}=0^{\circ }\quad \Longrightarrow \quad \sin \theta _{y}=0;\quad \cos \theta _{y}=1\quad \Longrightarrow \quad r_{02}=0;\quad r_{12}=0;\quad r_{22}=1;\quad r_{20}=0;\quad r_{21}=0$

${\begin{alignedat}{8}-r_{01}&=r_{10}&&=\sin \theta _{z_{1}}\cdot \cos \theta _{z_{2}}+\cos \theta _{z_{1}}\cdot \sin \theta _{z_{2}}\quad &&\Longrightarrow \quad -r_{01}&&=r_{10}&&=\sin(\theta _{z_{1}}+\theta _{z_{2}})\\r_{00}&=r_{11}&&=\cos \theta _{z_{1}}\cdot \cos \theta _{z_{2}}-\sin \theta _{z_{1}}\cdot \sin \theta _{z_{2}}\quad &&\Longrightarrow \quad r_{00}&&=r_{11}&&=\cos(\theta _{z_{1}}+\theta _{z_{2}})\end{alignedat}}$

$\Longrightarrow \quad \theta _{z_{1}}+\theta _{z_{2}}={\text{atan2}}(-r_{01},r_{00})={\text{atan2}}(r_{10},r_{11})\quad {\text{mit}}\quad \theta ={\text{atan2}}(\sin \theta ,\cos \theta )\quad {\text{Lösung mit einem Freiheitsgrad,}}\ \theta _{x}\ {\text{oder}}\ \theta _{z}\ {\text{ist frei wählbar}}$

${\text{(3) für}}\quad \theta _{y}=180^{\circ }\quad \Longrightarrow \quad \sin \theta _{y}=0;\quad \cos \theta _{y}=-1\quad \Longrightarrow \quad r_{02}=0;\quad r_{12}=0;\quad r_{22}=-1;\quad r_{20}=0;\quad r_{21}=0$

${\begin{alignedat}{8}-r_{01}&=-r_{10}&&=\sin \theta _{z_{1}}\cdot \cos \theta _{z_{2}}-\cos \theta _{z_{1}}\cdot \sin \theta _{z_{2}}\quad &&\Longrightarrow \quad -r_{01}&&=-r_{10}&&=\sin(\theta _{z_{1}}-\theta _{z_{2}})\\-r_{00}&=r_{11}&&=\cos \theta _{z_{1}}\cdot \cos \theta _{z_{2}}+\sin \theta _{z_{1}}\cdot \sin \theta _{z_{2}}\quad &&\Longrightarrow \quad r_{00}&&=r_{11}&&=\cos(\theta _{z_{1}}-\theta _{z_{2}})\end{alignedat}}$

$\Longrightarrow \quad \theta _{z_{1}}-\theta _{z_{2}}={\text{atan2}}(-r_{01},r_{00})={\text{atan2}}(-r_{10},r_{11})\quad {\text{mit}}\quad \theta ={\text{atan2}}(\sin \theta ,\cos \theta )\quad {\text{Lösung mit einem Freiheitsgrad,}}\ \theta _{x}\ {\text{oder}}\ \theta _{z}\ {\text{ist frei wählbar}}$

Imperial Measures

$(\,(\,\underbrace {\underbrace {\underbrace {\underbrace {\underbrace {\underbrace {\underbrace {25.4{\text{ mm}}} _{\text{= 1 inch}}\;\times \;2\cdot 2\cdot 3\;=304.8{\text{ mm}}} _{\text{= 1 foot = 12 inch}}\,)\;\times \;3\;=914.4{\text{ mm}}} _{\text{= 1 yard = 3 foot = 36 inch}}\,)\;\times \;5.5} _{\text{= 1 rod = 5.5 yard}}\;\times \;2\cdot 2} _{\text{= 1 chain = 4 rod = 22 yard}}\;\times \;2\cdot 5} _{\text{= 1 furlong = 10 chain = 220 yard}}\;\times \;2\cdot 2\cdot 2\;=1{,}609{,}344{\text{ mm}}} _{\text{= 1 mile = 8 furlong = 80 chain = 1,760 yard = 5,280 foot = 63,360 inch}}$

Continuously Variable Transmission

${\begin{alignedat}{4}{\text{with}}\quad &i_{12}={\frac {\omega _{1}}{\omega _{2}}}\ {\text{and}}\ \omega _{3}=0\ &&\Longrightarrow \ i_{23}=1-{\frac {1}{i_{12}}}\ ;\ \ {\text{with}}\ i_{23}={\frac {\omega _{2}}{\omega _{3}}}\ {\text{and}}\ \omega _{1}=0\qquad {\text{and}}\qquad i_{31}={\frac {1}{1-i_{12}}}\ ;\ \ {\text{with}}\ i_{31}={\frac {\omega _{3}}{\omega _{1}}}\ {\text{and}}\ \omega _{2}=0\\{\text{with}}\quad &i_{12}={\frac {n}{m}}&&\Longrightarrow \ i_{23}={\frac {n-m}{n}}\qquad {\text{and}}\qquad \ i_{31}={\frac {m}{m-n}}\\&\qquad \ i_{12}\ \cdot \ i_{23}\ \cdot \ i_{31}&&=\quad i_{12}\ \cdot \ {\frac {-(1-i_{12})}{i_{12}}}\ \cdot \ {\frac {1}{1-i_{12}}}\quad =\quad {\frac {n}{m}}\ \cdot \ {\frac {n-m}{n}}\ \cdot \ {\frac {m}{-(n-m)}}\quad =\quad -1\end{alignedat}}$

Geometrische Reihe

Konvergenzkriterium

Konvergenz der geometrischen Reihe für $\textstyle \sum _{k=0}^{\infty }({\tfrac {1}{2}})^{k}=2$
als Flächeninhalte dargestellt

Konvergenz der geometrischen Reihe $\textstyle {\tfrac {1}{2}}\sum _{k=0}^{\infty }({\tfrac {1}{2}})^{k}=1$
auf einer Zahlengeraden dargestellt

Für $|q|<1$ konvergiert die geometrische Reihe. Es gilt in diesem Fall

$\sum _{k=0}^{\infty }q^{k}=1+q+q^{2}+\dots ={\frac {1}{1-q}}$

Die Herleitung erfolgt über eine Betrachtung der Differenz von Partialsummen der Reihe

${\begin{alignedat}{4}s_{n}\ &&:=&&\sum _{k=0}^{n}q^{k}&=\ 1+q\;\;+&&\,q^{2}+q^{3}+\dots +q^{n},\\q\cdot s_{n}&&=&&q\cdot \sum _{k=0}^{n}q^{k}&=\ q+q^{2}\,+&&\,q^{3}+q^{4}+\dots +q^{n+1},\\s_{n}-q\cdot s_{n}&&=&&\ (1-q)\cdot s_{n}&=\ 1-q^{n+1},&&\\s_{n}&&=&&{\frac {1-q^{n+1}}{1-q}}\,&\end{alignedat}}$

Es ist $q^{n}$ für $|q|<1$ eine Nullfolge, also gilt

$\lim _{n\to \infty }s_{n}=\lim _{n\to \infty }{\frac {1-q^{n+1}}{1-q}}={\frac {1-0}{1-q}}={\frac {1}{1-q}}$.

Es ist also $|q|<1$ hinreichend für die Konvergenz der geometrischen Reihe. Zugleich ist dies jedoch auch notwendig: Für $|q|\geq 1$ folgt die Divergenz der Reihe $\textstyle \sum _{n=0}^{\infty }q^{n}$ aus dem Nullfolgenkriterium, da $q^{n}$ in diesem Fall für $n\to \infty$ nicht gegen 0 strebt.

Ein Quotient $q$ mit $|q|\geq 1$ ergibt eine divergente geometrische Reihe, z. B. für $q=2$ und Startwert $1$

$1,~1+2,~1+2+4,~1+2+4+8,\dots$ zusammengefasst also $1,~3,~7,~15,\dots$.

Im Falle der hier abgebildeten Zweierpotenzen erscheinen stets die Mersenneschen Zahlen $s_{n}=2^{n+1}-1$ als Werte der Summe.