<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] The second (finite) series is a particularization of the first (with a shifted index):
$$
1 + x + x^{2} + x^{3} + \cdots + x^{n} + O(x^{n+1})= \frac{1}{1 - x}
$$
means: There exists constant $C > 0$ and $\delta > 0$ such that if $|x| < \delta$, then
$$
\left|\frac{1}{1 - x} - (1 + x + x^{2} + x^{3} + \cdots + x^{n})\right| \leq C|x|^{n+1}.
\tag{1}
$$
By contrast, the second (with $n$ replaced by $n + 1$) reads
$$
1 + r + r^{2} + r^{3} + \cdots + r^{n}
= \frac{r^{n+1} - 1}{r - 1}
= \frac{1 - r^{n+1}}{1 - r}
= \frac{1}{1 - r} - \frac{r^{n+1}}{1 - r}.
\tag{2}
$$
Equation (2) gives the specific form of the $O(x^{n+1})$, namely $\dfrac{x^{n+1}}{1 - x}$.
</s>