The value of \(\lim _{n \rightarrow \infty} \frac{\pi}{n}\left[\sin \frac{\pi}{n}+\sin \frac{2 \pi}{n}+\ldots .+\sin \frac{(n-1) \pi}{n}\right]\) is
Step-by-step Solution:
\(r^{\text {th }}\) term of the given series is \[\] \(\sin \left(\frac{r \pi}{n}\right) \frac{\pi}{n}\) \[\] Sum of the series is given by \(\sum_{r=1}^{n-1} \frac{\pi}{n} \sin \left(\frac{r \pi}{n}\right)\) \[\] Putting \(\frac{r}{n}=x \Rightarrow \frac{1}{n}=d x\) \[\] Thus the sum is \(\int_{0}^{1} \pi \sin (\pi x) d x=2 . \quad\) Choice (C)\[\]