What is the solution for the following sum? $\sum_{i=1}^n i^{-\alpha}$ for $-2\leq i\leq 0$.

For example if we have $\alpha =-1 $ we know it is the harmonic series thus $=\log n$ and for $\alpha=-2$ it is $\Theta(1)$. What if $\alpha$ ranges? Is there any more general formula?

How about if $\alpha>1$? Can we say it is $\Theta(n^{1+\alpha})$? I can easily be verified by Gauss trick that $O(n^{1+\alpha})$ holds. How about the lower bound?