Sequences & Series
Sum of Infinite AGP
Grade 11
Question:
<p>Find the sum of the infinite series \(1 - 3x + 5x^2 - 7x^3 + \cdots \infty\) when \(0 < |x| < 1\).</p>
<p>\(\dfrac{1-x}{(1+x)^2}\)</p>
<p>\(\dfrac{1+x}{(1-x)^2}\)</p>
<p>\(\dfrac{1}{(1+x)^2}\)</p>
<p>\(\dfrac{1-x}{(1-x)^2}\)</p>
Step-by-Step Solution
Key Concept: Recognize the series as an arithmetic-geometric progression where coefficients are (2n-1) with alternating signs. Use the derivative of geometric series or multiply by (1+x) to find a telescoping relationship.
<p><strong>Step 1:</strong> Write the series as S = 1 - 3x + 5x² - 7x³ + ... = ∑(2n-1)(-x)ⁿ⁻¹ for n=1 to ∞</p><p><strong>Step 2:</strong> Let S = 1 - 3x + 5x² - 7x³ + ...</p><p><strong>Step 3:</strong> Multiply by (-x): -xS = -x + 3x² - 5x³ + 7x⁴ - ...</p><p><strong>Step 4:</strong> Add S + (-xS) = S(1+x):</p><p>S(1+x) = 1 - 2x + 2x² - 2x³ + 2x⁴ - ... = 1 - 2x(1 - x + x² - x³ + ...)</p><p><strong>Step 5:</strong> The geometric series: 1 - x + x² - x³ + ... = 1/(1+x) for |x| < 1</p><p><strong>Step 6:</strong> S(1+x) = 1 - 2x · 1/(1+x) = 1 - 2x/(1+x) = (1+x-2x)/(1+x) = (1-x)/(1+x)</p><p><strong>Step 7:</strong> S = (1-x)/(1+x)²</p><p>∴ Answer: A</p>
Correct Answer: A