I'm preparing for an exam, and one of the review problems is to sort functions by order of growth, and this was the only summation in it. I know that
$$\sum \limits_{i=1}^n i^2 = \frac{n(n+1)(2n+1)}{6},$$
But what if I did not know the closed form? How, then, would I prove
$$\sum \limits_{i=1}^n i^2 \in \Theta (n^3).$$