Partialbruchzerlegung/Z/Potenz im Nenner/Maximal/Aufgabe

Zeige, dass für Zahlen ${}n,r\in \mathbb {N} _{+}$ die Gleichheit

{}{\frac {n^{r}-1}{n^{r}}}={\frac {n-1}{n}}+{\frac {n-1}{n^{2}}}+\cdots +{\frac {n-1}{n^{r-1}}}+{\frac {n-1}{n^{r}}}\,

gilt.

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