I'm stuck with this exercise: I have to find for which $x$ the estimate $\displaystyle\sum\limits_{i=0}^{n}x^i=O(n)$ holds.
It seems intuitive to me that this must be the case for all $x \in (0,1)$ but proving this seems to be beyond my abilities.
I tried some different approaches like the usual $\displaystyle \lim\limits_{x\to\infty}\frac{f(x)}{g(x)} = \text{some finite value}$ with $f(x)$ the formula for the partial sums. I tried the same thing with l'Hôpital's rule. I also tried to argue that the highest exponent of the sum must be $x^n$ and therefore I can just say that this holds for all $0 < x < 1$, but that doesn't seem very convincing to me.
I am out of ideas how to solve this problem and everything I try feels wrong to me, I hope someone in this community can help me.