Question

Evaluate \[ \lim_{n \to \infty} \frac{1}{2n} \left( \sin \frac{\pi}{2n} + \sin \frac{\pi}{n} + \sin \frac{2\pi}{2n} + \dots \right) = ? \]

• A. 1
• B. 0
• C. 4
• D. 3

Hint:

Recognize sums like this as Riemann sums for definite integrals.

Answer 1

The Correct Option is A

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.