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The expression is $\frac{r^N - \left( r-\epsilon \right )^N}{r^N}=1 - \left ( 1- \frac{\epsilon}{r} \right )^N$.

I understand where the first $1$ comes from, but where does the $\left ( 1- \frac{\epsilon}{r} \right )^N$ come from? It looks like $(r-\epsilon)^N $ can simply be divided by r. Why does this work?

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$$ \frac{r^N - \left( r-\epsilon \right )^N}{r^N}=\frac{r^N}{r^N} - \frac{(r-\epsilon)^N}{r^N}=1-\left ( \frac{r-\epsilon}{r} \right )^N=1-\left ( 1- \frac{\epsilon}{r} \right )^N. $$

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$1-(1-\frac{\epsilon}{r})^N = (1-(1-\frac{\epsilon}{r})^N)\cdot 1 = (1-(1-\frac{\epsilon}{r})^N)\cdot \frac{r^N}{r^N}$

$=\frac{r^N-r^N(1-\frac{\epsilon}{r})^N}{r^N}=\frac{r^N-(r(1-\frac{\epsilon}{r}))^N}{r^N}=\frac{r^N - \left( r-\epsilon \right )^N}{r^N}$

This is just an application of "multiplying by one" and is true for all numbers where $r\neq 0$, no condition on positiveness required.

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