Question:
Evaluate
\[
\lim_{n \to \infty} \frac{1}{2n} \left( \sin \frac{\pi}{2n} + \sin \frac{\pi}{n} + \sin \frac{2\pi}{2n} + \dots \right) = ?
\]
Evaluate
\[
\lim_{n \to \infty} \frac{1}{2n} \left( \sin \frac{\pi}{2n} + \sin \frac{\pi}{n} + \sin \frac{2\pi}{2n} + \dots \right) = ?
\]
Show Hint
Recognize sums like this as Riemann sums for definite integrals.
Updated On: Jun 6, 2025
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The Correct Option is A
Solution and Explanation
This is a Riemann sum for the integral of \(\sin(x)\) over the interval \( [0, \pi] \). As \(n \to \infty\), the sum converges to:
\[
\int_0^\pi \sin(x) \, dx.
\]
The result of this definite integral is \(1\), so the correct answer is 1.
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