The value of \(\lim _{n \rightarrow \infty} \frac{1}{n}\left\{\sec ^{2} \frac{\pi}{4 n}+\sec ^{2} \frac{2 \pi}{4 n}+\ldots \ldots+\sec ^{2} \frac{n \pi}{4 n}\right\}\) is

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The value of \(\lim _{n \rightarrow \infty} \frac{1}{n}\left\{\sec ^{2} \frac{\pi}{4 n}+\sec ^{2} \frac{2 \pi}{4 n}+\ldots \ldots+\sec ^{2} \frac{n \pi}{4 n}\right\}\) is

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kritika · Answer 1 · 2021-12-27T06:59:23+0000

Ans: (C)
Hint : Required limit \(=\operatorname{It}_{n \rightarrow \infty} \sum_{r=1}^{n} \frac{1}{n} \sec ^{2}\left(\frac{r \pi}{4 n}\right)=\int_{0}^{1} \sec ^{2}\left(\frac{\pi x}{4}\right) d x=\left.\frac{4}{\pi} \tan \left(\frac{\pi x}{4}\right)\right|_{0} ^{1}=\frac{4}{\pi}\)

The value of \(\lim _{n \rightarrow \infty} \frac{1}{n}\left\{\sec ^{2} \frac{\pi}{4 n}+\sec ^{2} \frac{2 \pi}{4 n}+\ldots \ldots+\sec ^{2} \frac{n \pi}{4 n}\right\}\) is

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