<p>Evaluate the following limit: <br> 8. \(\lim_{n \to \infty} \frac{1}{n}\left(\sin\frac{\pi}{2n} + \sin\frac{2\pi}{2n} + \sin\frac{3\pi}{2n} + \cdots + \sin\frac{n\pi}{2n}\right)\)</p>
Step-by-Step Solution
Key Concept: Recognize this sum as a Riemann sum for the integral ∫₀^(π/2) sin(x)dx by writing the sum as (1/n)∑sin(kπ/2n) where kπ/2n are partition points on [0, π/2].
<p><strong>Step 1:</strong> Rewrite the sum as a Riemann sum:</p><p>Let S = (1/n)∑(k=1 to n) sin(kπ/2n)</p><p>This can be written as: S = ∑(k=1 to n) sin(kπ/2n) · (1/n)</p><p><strong>Step 2:</strong> Identify the Riemann sum structure:</p><p>Set Δx = (π/2)/n and x_k = kπ/2n. Then:</p><p>S = ∑(k=1 to n) sin(x_k) · Δx</p><p>This is a right Riemann sum for f(x) = sin(x) on interval [0, π/2] with partition width Δx = π/(2n)</p><p><strong>Step 3:</strong> Apply the definition of definite integral:</p><p>lim(n→∞) S = ∫₀^(π/2) sin(x)dx</p><p><strong>Step 4:</strong> Evaluate the definite integral:</p><p>∫₀^(π/2) sin(x)dx = [-cos(x)]₀^(π/2) = -cos(π/2) + cos(0) = 0 + 1 = 1</p><p>∴ Answer: <strong>1</strong></p>
Correct Answer: 1