Finding out the limit: $\lim_{n \to \infty}\frac{1}{n}\sum_{k=0}^{n}\sqrt {k(n-k)}$ Find out the limit of the sum   $\lim \frac{1}{n}\sum_{k=0}^{n}\sqrt {k(n-k)}$ as $n$ tends to $\infty.$

Since $x-x^2$ has a peak at $x=\frac{1}{2}$ and decreases monotonically to $0$ as $x \downarrow 0$ or $x \uparrow 1$, we see $\frac{1}{n}\sum_{k=0}^n \sqrt{k(n-k)} = \sum_{k=1}^{n-1} \sqrt{\frac{k}{n}(1-\frac{k}{n})} \ge \sum_{k=1}^{n-1} \sqrt{\frac{1}{n}(1-\frac{1}{n})} = (n-1)\sqrt{\frac{1}{n}-\frac{1}{n^2}} = \sqrt{\frac{(n-1)^2}{n}-\frac{(n-1)^2}{n^2}} \sim \sqrt{n}$ which diverges
