What is $1+x+x^2+x^3...$? What is the difference between these two series? 
$$
\begin{align}
1+x+x^2+x^3+...+x^n+\mathcal O(x^{n+1})&=\frac{1}{1-x}\\
\\
1+r+r^2+r^3+...+r^{n-1}&=\frac{r^n-1}{r-1}\\
\end{align}
$$
I can't wrap my head around it; they both start with $1+x+x^2+x^3...$? Of course the first is the maclaurin series. But other than that, why don't they both yield the same result?
Thanks!

Beware that "$=$" is not a symmetric relation in the presence of asymptotic notation. So while it is true that
$$\frac{1}{1-x} = 1+x+x^2+x^3+...+x^n+O(x^{n+1})$$
as $x\to 0$, we do not usually say that
$$1+x+x^2+x^3+...+x^n+O(x^{n+1})=\frac{1}{1-x}$$
because the left-hand side of that can be a lot of things other than $\frac{1}{1-x}$.
Some consider the conventional use of $=$ in these contexts to be misguided and misleading, and would rather we write
$$\frac{1}{1-x} \in 1+x+x^2+x^3+...+x^n+\mathcal O(x^{n+1})$$
whose right-hand side is interpreted as a set of functions.
