29.50. Calculate the limit $\lim _{n \rightarrow \infty} \sum_{k=0}^{n-1} \frac{\sqrt{n^{2}-k^{2}}}{n^{2}}$.
29.50. Answer: $\pi / 4$. The considered sum can be written as $\sum_{k=0}^{n-1} \frac{1}{n} \sqrt{1-\left(\frac{k}{n}\right)^{2}}$. As $n \rightarrow \infty$, this sum tends to $\kappa \int_{0}^{1} \sqrt{1-x^{2}} d x=\frac{\pi}{4}$.