<p>If \(1 - \dfrac{1}{3} + \dfrac{1}{5} - \dfrac{1}{7} + \dfrac{1}{9} - \dfrac{1}{11} + \cdots = \dfrac{\pi}{4}\), then value of \(\dfrac{1}{1 \times 3} + \dfrac{1}{5 \times 7} + \dfrac{1}{9 \times 11} + \cdots\) is</p>
Step-by-Step Solution
Key Concept: The given series is Leibniz's formula for π/4. The target series can be obtained by pairing and regrouping terms using partial fractions: 1/(4k-3)(4k-1) = (1/2)[1/(4k-3) - 1/(4k-1)], which relates directly to alternating groups in the original series.
<p><strong>Step 1:</strong> Recognize the given series as Leibniz's formula:</p><p>∑((-1)^n)/(2n+1) = 1 - 1/3 + 1/5 - 1/7 + 1/9 - 1/11 + ... = π/4</p><p><strong>Step 2:</strong> Group the alternating series in pairs: (1 - 1/3) + (1/5 - 1/7) + (1/9 - 1/11) + ...</p><p>= 2/3 + 2/35 + 2/99 + ... = π/4</p><p><strong>Step 3:</strong> Simplify each term using partial fractions. For the general term:</p><p>1/(4k-3) - 1/(4k-1) = 2/[(4k-3)(4k-1)]</p><p><strong>Step 4:</strong> Therefore: 2/3 + 2/35 + 2/99 + ... = 2[1/(1×3) + 1/(5×7) + 1/(9×11) + ...] = π/4</p><p><strong>Step 5:</strong> Divide both sides by 2:</p><p>1/(1×3) + 1/(5×7) + 1/(9×11) + ... = π/8</p><p>∴ Answer: <strong>π/8</strong></p>
Correct Answer: A