<p>Let \(E = \dfrac{1}{1^2}+\dfrac{1}{2^2}+\dfrac{1}{3^2}+\cdots\). Then,</p>
Step-by-Step Solution
Key Concept: The infinite series ∑(1/n²) converges to π²/6, a fundamental result proven using Fourier series or the Basel problem. Recognizing this standard series is essential for JEE-level problems.
<p><strong>Step 1:</strong> Identify the series structure: E = 1/1² + 1/2² + 1/3² + ... = Σ(1/n²) for n=1 to ∞</p><p><strong>Step 2:</strong> Recognize this as the Basel problem, a classical result in mathematics.</p><p><strong>Step 3:</strong> Apply the standard result that this p-series with p=2 (p>1) converges. The exact sum is proven via Fourier series of f(x)=x on [-π,π].</p><p><strong>Step 4:</strong> The closed form is E = π²/6 ≈ 1.6449</p><p><strong>Key insight:</strong> This is one of the most important standard series in calculus. Euler famously solved this in 1734, proving it equals π²/6.</p><p>∴ Answer: A (The series converges to π²/6)</p>
Correct Answer: A