calculus - Is the limit finite? (corrected)

I need to find $r>0$ for which the following limit is finite

$$\lim_{n \rightarrow \infty} \sum_{k=1}^{n^2} \frac{n^{r-1}}{n^r+k^r}$$

I get inconclusiveness using the ratio test. The root test does not seem to help me. Does it converge to zero to for $r \in \mathbb Z^+$?

Any ideas?