AMC 12 2016b Problem 21

Let $ABCD$ be a unit square. Let $Q_1$ be the midpoint of $\overline{CD}$. For $i=1,2,\dots,$ 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 $$\sum_{i=1}^{\infty} \text{Area of } \triangle DQ_i P_i ?$$

\textbf{(A)}\ \frac{1}{6} \textbf{(B)}\ \frac{1}{4} \textbf{(C)}\ \frac{1}{3} \textbf{(D)}\ \frac{1}{2} \textbf{(E)}\ 1