<s>[INST] 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!
 [/INST] Hint:
$$1+x+x^2+\dots+x^n=\frac{x^{n+1}-1}{x-1}$$
$$\frac{x^{n+1}-1}{x-1}=\frac{1-x^{n+1}}{1-x}=\frac1{1-x}-\frac{x^{n+1}}{1-x}=\frac1{1-x}+\mathcal O(x^{n+1})$$
which holds as $x\to0$, since $1-x\to1$.
Move $\mathcal O(x^{n+1})$ to the other side and you'll get
$$1+x+x^2+\dots+x^n+\mathcal O(x^{n+1})=\frac1{1-x}$$
</s>