Hello i have to solve the following problem find an $\displaystyle f(k)$ where $\displaystyle S_k=\theta(f(k))$ where $\displaystyle S_k =\sum_{n=1}^{k^2-1} \sqrt{n}$
I tried first of all to calculate or "limit" my sum using integral rule so i came up with $\displaystyle \frac{2(k^2-1)^{3/2}}{3} \leq S_k \leq \frac{2(k^3-1)}{3}$
but after that i am in a dead end as i do not know $\displaystyle S_k$ so i can not simplify anything can anyone help how to proceed this problem ? thanks