$\lim _{n \rightarrow \infty}\left[\frac{1^k+2^k+3^k+\ldots+n^k}{n^{k+1}}\right]=$

$\lim _{n \rightarrow \infty}\left(\frac{1}{1^2+n^2}+\frac{2}{2^2+n^2}+\frac{3}{3^2+n^2}+\ldots+\frac{n}{n^2+n^2}\right)=$

$\lim _{n \rightarrow \infty} n^4\left[\frac{1}{n^5}+\frac{1}{\left(n^2+1\right)^{\frac{5}{2}}}+\frac{1}{\left(n^2+4\right)^{\frac{5}{2}}}+\frac{1}{\left(n^2+9\right)^{\frac{5}{2}}}+\ldots+\right]=$

$\lim _{n}$ ${\rightarrow \infty}\left[\left(1+\frac{1}{n^2}\right)\left(1+\frac{4}{n^2}\right)\left(1+\frac{9}{n^2}\right) \ldots \left(1+\frac{n^2}{n^2}\right)\right]^{1 / n}=$

$\lim_{n \to \infty} \frac{\sqrt{1} + \sqrt{2} + \dots + \sqrt{n}}{n^{\frac{3}{2}}} =$

$\lim _{n}$ ${\rightarrow \infty}\left(\frac{n}{n^2+1^2}+\frac{n}{n^2+2^2}+\frac{n}{n^2+3^2}+\ldots+\frac{n}{n^2+(2n)^2}\right)=$