I'm reading the derivation of the geometric sum formula here: http://www.purplemath.com/modules/series7.htm
It says that the polynomial property: $$\frac{x^n-1}{x-1} = 1+x+x^2+...+x^{n-1}$$ (i.e. any polynomial of the form $x^n-1$ is divisible by $x-1$)
Applies to the geometric series $1+r+r^2+...+r^{n-1}$ producing: $$\frac{1-r^n}{1-r}$$ where the subtraction has been reversed.
I can't see why this reversing is legal.
It's exactly the same, because $$ \frac{r^n-1}{r-1}=\frac{-(1-r^n)}{-(1-r)}=\frac{1-r^n}{1-r} $$ Why is it written this way? Because, if $|r|<1$, then $$ \lim_{n\to\infty}r^n=0 $$ and so $$ \lim_{n\to\infty}\frac{1-r^n}{1-r}=\frac{1-0}{1-r}=\frac{1}{1-r} $$ Of course it would be the same writing $$ \lim_{n\to\infty}\frac{r^n-1}{r-1}=\frac{0-1}{r-1}= \frac{-1}{-(1-r)}=\frac{1}{1-r} $$ Just a matter of opinion about what's the clearer presentation.