Kunnskapsgnist
En rekke $\sum a_n$ er geometrisk dersom hvert ledd er et multiplum av det forrige, dvs. $a_{n+1} = \textcolor{red}{r} a_n$:
$$\textcolor{blue}{a} + \textcolor{blue}{a} \cdot \textcolor{red}{r} + \textcolor{blue}{a} \cdot \textcolor{red}{r} ^2 + \textcolor{blue}{a} \cdot \textcolor{red}{r} ^3 + \cdots = \sum_{n=1}^{\infty} \textcolor{blue}{a} \textcolor{red}{r} ^{n-1} $$$\textcolor{red}{r}$ kalles kvotienten.
$|\textcolor{red}{r}| < 1$: Rekken konvergerer og summen er:
$$S = \sum_{n = 1}^{\infty} \textcolor{blue}{a} \textcolor{red}{r} ^{n-1}= \frac{\textcolor{blue}{a}}{1-\textcolor{red}{r}}$$$|\textcolor{red}{r}| \ge 1$: Rekken divergerer.
$$\textcolor{white}{\sum_{n = 1}^{\infty}} $$
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