Question:

<div class="solution"><p><strong>Key Idea:</strong> Factor denominator: $r^4+2r^3+r^2+1=1+(r^2+r)^2$. Numerator $=(r+1)^2-r^2$.</p><p><strong>Step 1:</strong> $T_r=\tan^{-1}(r+1)^2-\tan^{-1}r^2$.</p><p><strong>Step 2:</strong> Telescoping: sum $=\tan^{-1}(n+1)^2-\tan^{-1}(1)\xrightarrow{n\to\infty}\pi/2-\pi/4=\pi/4$.</p><p><strong>Answer: (A) $\pi/4$</strong></p><div class="trap-box"><strong>Trap:</strong> Ignoring the subtracted initial term $\tan^{-1}(1)=\pi/4$.<div class="key-concept"><strong>Key Concept:</strong> Quartic denominator hides $1+(r^2+r)^2$ -- numerator is difference of squares