3. A If the bug walks infinitely, the algebraic sum of its horizontal movement, i.e., the limit of the geometric series $1-\frac{1}{4}+\frac{1}{16}-\cdots=$ $\frac{1}{1-\left(-\frac{1}{4}\right)}=\frac{4}{5}$. Similarly, the algebraic sum of its vertical movement approaches $\frac{1}{2}-\frac{1}{8}+\frac{1}{32}-\cdots=\frac{2}{5}$.
3. A bug (ignoring its size) starts from the origin of the coordinate plane, first it moves one unit to the right to $(1,0)$, then it rotates $90^{\circ}$ counterclockwise and walks $\frac{1}{2}$ unit to $\left(1, \frac{1}{2}\right)$. If it continues in this manner, rotating $90^{\circ}$ counterclockwise each time and walking half the distance of the previous step, which of the following points is closest to the bug?
A. $\left(\frac{4}{5}, \frac{2}{5}\right)$
B. $\left(\frac{2}{3}, \frac{4}{5}\right)$
C. $\left(\frac{2}{3}, \frac{1}{3}\right)$
D. $\left(\frac{2}{5}, \frac{4}{5}\right)$