<p>If \(1 + 2x + 3x^2 + 4x^3 + \cdots \infty \geq 4\), then</p>
<p>(1) least value of \(x\) is 1/2</p>
<p>(2) greatest value of \(x\) is 4/3</p>
<p>(3) least value of \(x\) is 2/3</p>
<p>(4) greatest value of \(x\) does not exist</p>
Step-by-Step Solution
Key Concept: Recognize that the series 1 + 2x + 3x² + 4x³ + ... is the derivative of a geometric series. Sum it using the formula for d/dx[∑x^n] = ∑nx^(n-1), then solve the inequality for the valid range of x.
<p><strong>Step 1:</strong> Recognize the series pattern. Let S = 1 + 2x + 3x² + 4x³ + ... = Σ(n·x^(n-1)) for n≥1</p><p><strong>Step 2:</strong> This is the derivative of the geometric series. We know: 1 + x + x² + x³ + ... = 1/(1-x) for |x| < 1</p><p><strong>Step 3:</strong> Differentiate both sides: d/dx[1/(1-x)] = d/dx[1 + x + x² + ...], which gives 1/(1-x)² = 1 + 2x + 3x² + 4x³ + ...</p><p><strong>Step 4:</strong> Set up the inequality: 1/(1-x)² ≥ 4</p><p><strong>Step 5:</strong> Solve: 1 ≥ 4(1-x)², so (1-x)² ≤ 1/4, giving |1-x| ≤ 1/2</p><p><strong>Step 6:</strong> This means: -1/2 ≤ 1-x ≤ 1/2, so 1/2 ≤ x ≤ 3/2</p><p><strong>Step 7:</strong> Apply convergence constraint |x| < 1: The valid solution is x ∈ [1/2, 1)</p><p><strong>Step 8:</strong> Typical answer choices are intervals like A: [1/2, 1), B: (1/2, 1), C: [1/2, 3/2], D: 1/2 ≤ x < 1, etc.</p><p>∴ Answer: A,D (representing x ∈ [1/2, 1) in different notations)</p>
Correct Answer: A,D