Take the 2-minute tour ×  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.  share|improve this
question add comment  1 Answer  up vote 5 down vote accepted
Hint: If $0\leq x\leq 1 $ then $x^i\leq 1$ so that
$$\left|\sum_{i=0}^n x^i\right|\leq n+1.$$  For $x>1$ notice
this is a geo