The value of lim(n→∞) Σ(r=1 to n) (r/n²) sin(rπ/2n) is:

The value of lim(n→∞) Σ(r=1 to n) (r/n²) sin(rπ/2n) is:

• A. 2/π
• B. 4/π²
• C. π/2
• D. 1/π

Answer: B) 4/π²

Explanation: Rewriting as lim(n→∞) (1/n) Σ (r/n) sin(rπ/2n). Let x = r/n, dx = 1/n. As r goes 1 to n, x goes 0 to 1. Integral = ∫₀¹ x sin(πx/2) dx. Integrate by parts: u = x, dv = sin(πx/2) dx. v = −(2/π) cos(πx/2). Integral = [−x(2/π) cos(πx/2)]₀¹ + ∫₀¹ (2/π) cos(πx/2) dx = −(2/π)(0 − 0) + (2/π)[(2/π) sin(πx/2)]₀¹ = (4/π²)[1 − 0] = 4/π².