<p>If \(\dfrac{1}{1^2} + \dfrac{1}{2^2} + \dfrac{1}{3^2} + \cdots\) to \(\infty = \dfrac{\pi^2}{6}\), then \(\dfrac{1}{1^2} + \dfrac{1}{3^2} + \dfrac{1}{5^2} + \cdots\) equals</p>
Step-by-Step Solution
Key Concept: Separate the Basel problem sum into odd and even indexed terms: the sum of all reciprocal squares equals the sum of odd-indexed terms plus the sum of even-indexed terms. The even-indexed terms form a pattern that can be expressed in terms of the original series.
<p><strong>Step 1:</strong> Let S = 1/1² + 1/2² + 1/3² + ... = π²/6</p><p><strong>Step 2:</strong> Decompose S into odd and even indexed terms:</p><p>S = (1/1² + 1/3² + 1/5² + ...) + (1/2² + 1/4² + 1/6² + ...)</p><p><strong>Step 3:</strong> The even-indexed sum = 1/2² + 1/4² + 1/6² + ... = 1/4(1/1² + 1/2² + 1/3² + ...) = (1/4) · π²/6</p><p><strong>Step 4:</strong> Let the odd-indexed sum = O. Then:</p><p>π²/6 = O + (1/4)(π²/6)</p><p>π²/6 = O + π²/24</p><p>O = π²/6 - π²/24 = 4π²/24 - π²/24 = 3π²/24 = <strong>π²/8</strong></p><p>∴ Answer: A</p>
Correct Answer: A