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