29.52. Compute $\lim _{n \rightarrow \infty} \sum_{k=0}^{n-1} \frac{1}{\sqrt{n^{2}-k^{2}}}$.

See also problem 25.42.

### 29.10. Identities
29.52. Answer: $\pi / 2$. The considered sum can be written as $\sum_{k=0}^{n-1} \frac{1}{n} \frac{1}{\sqrt{1-(k / n)^{2}}}$. This sum tends to $\int_{0}^{1} \frac{d x}{\sqrt{1-x^{2}}}=$ $=\left.\arcsin x\right|_{0} ^{1}=\frac{\pi}{2}$.