Is the limit finite? (corrected)

Question

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?

Answer 1

Hint: rewrite this as $$\lim_{n\to\infty} \frac 1n \sum_{k=1}^{n^2} \frac1{1 + \left(\frac kn\right)^r} $$

Answer 2

For $r\geq1$ the partial sums, counted to $n$ behave as $(1/n)$ and the longer ones are bigger, hence it diverges.

Answer 3

We first observe that $$ \sum_{k=1}^{n^2} \frac{n^{r-1}}{n^r+k^r}\ge \sum_{k=1}^{n^2} \frac{n^{r-1}}{n^r+n^r}=n^2\cdot\frac{1}{2n}=\frac{n}{2}\to \infty. $$ Therefore, for every $r\ge 0$, $$ \lim_{n\to\infty}\sum_{k=1}^{n^2} \frac{n^{r-1}}{n^r+k^r}=\infty. $$