<p>Let \(L = \lim_{n \to \infty} \dfrac{1}{n^3} \sum_{k=1}^{n} k^2 e^{\frac{k}{n}}\), then find the value of \(e - L\).</p>
Step-by-Step Solution
Key Concept: Convert the Riemann sum to a definite integral by recognizing that (1/n³)∑k²e^(k/n) = (1/n)∑(k/n)²e^(k/n) where k/n represents partition points. This transforms into ∫₀¹ x²eˣ dx after proper limiting analysis.
<p><strong>Step 1: Rewrite as Riemann sum</strong></p><p>L = lim(n→∞) (1/n³)∑(k=1 to n) k²e^(k/n) = lim(n→∞) (1/n)∑(k=1 to n) (k/n)²e^(k/n)</p><p>This is a Riemann sum with partition width Δx = 1/n and sample points xₖ = k/n.</p><p><strong>Step 2: Identify the integral</strong></p><p>L = ∫₀¹ x²eˣ dx</p><p><strong>Step 3: Evaluate using integration by parts</strong></p><p>Let u = x², dv = eˣ dx</p><p>Then du = 2x dx, v = eˣ</p><p>∫x²eˣ dx = x²eˣ - ∫2xeˣ dx</p><p><strong>Step 4: Apply integration by parts again</strong></p><p>For ∫2xeˣ dx: u = 2x, dv = eˣ dx</p><p>∫2xeˣ dx = 2xeˣ - ∫2eˣ dx = 2xeˣ - 2eˣ</p><p><strong>Step 5: Combine</strong></p><p>∫x²eˣ dx = x²eˣ - 2xeˣ + 2eˣ = eˣ(x² - 2x + 2)</p><p><strong>Step 6: Evaluate definite integral</strong></p><p>L = [eˣ(x² - 2x + 2)]₀¹ = e¹(1 - 2 + 2) - e⁰(0 - 0 + 2)</p><p>L = e(1) - 1(2) = e - 2</p><p><strong>Step 7: Find e - L</strong></p><p>e - L = e - (e - 2) = 2</p><p>∴ Answer: <strong>2</strong></p>
Correct Answer: 2