Let \(ABCD\) be a unit square. Let \(Q_1\) be the midpoint of \(\overline{CD}\). For \(i=1,2,\ldots,\) let \(P_i\) be the intersection of \(\overline{AQ_i}\) and \(\overline{BD}\), and let \(Q_{i+1}\) be the foot of the perpendicular from \(P_i\) to \(\overline{CD}\).

What is \(\sum\limits_{i=1}^{\infty} \text{Area of }\triangle DQ_iP_i\)?

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