Asymptotic growth of $\sum_{i=1}^n \frac{1}{i^\alpha}$? Let $0 < \alpha < 1$. Can somebody please explain why
$$\sum_{i=1}^n \frac{1}{i^\alpha} \sim n^{1-\alpha}$$
holds?

We have for $x\in [k,k+1)$ that $$(k+1)^{—\alpha}\leq x^{-\alpha}\leq k^{—\alpha},$$
and integrating this we get 
$$(k+1)^{—\alpha}\leq \frac 1{1-\alpha}((k+1)^{1-\alpha}-k^{\alpha})\leq k^{—\alpha}.$$
We get after summing and having used $(n+1)^{1-\alpha}-1\sim n^{1-\alpha}$, the equivalent $\frac{n^{1-\alpha}}{1-\alpha}$.
