Question:

<div class="solution"><p><strong>Key Idea:</strong> Factor denominator as $1+(n^2+n+1)(n^2-n+1)$.</p><p><strong>Step 1:</strong> <span class="math-block">$$T_n=\tan^{-1}\!\frac{(n^2+n+1)-(n^2-n+1)}{1+(n^2+n+1)(n^2-n+1)}=\tan^{-1}(n^2+n+1)-\tan^{-1}(n^2-n+1)$$</p><p><strong>Step 2:</strong> Telescoping: $\lim_{N\to\infty}[\tan^{-1}(N^2+N+1)-\tan^{-1}(1)]=\pi/2-\pi/4=\pi/4$.</p><p><strong>Answer: (B) \pi/4</strong></p><div class="trap-box"><strong>Trap:</strong> Expanding the quartic blindly kills the telescope. Spot the product of two neighboring quadratics.<div class="key-concept"><strong>Key Concept:</strong> Quartic denominators hide product of neighboring quadratics -- standard ITF telescope