Asymptotic growth of a function containing a sum
There is an explicit formula for $\sum_{i=0}^n i^3$, but even without it, you can estimate
$$
\int_0^n x^3 \, dx \leq \sum_{i=0}^n i^3 \leq \int_1^{n+1} x^3 \, dx.
$$
Since $\int x^3 \, dx = x^4/4$, this shows that the sum is very close to $n^4/4$, and in particular is $\Theta(n^4)$.

(The explicit formula states that the sum equals $n^2(n+1)^2/4$.)