An equilateral triangle is inscribed in circle $\mathrm { C } _ { 1 }$ with radius 1 unit as shown in the figure. Let $\mathrm { C } _ { 2 }$ be the circle passing through the midpoints of the sides of this triangle. The same operation is performed for circle $\mathrm { C } _ { 2 }$ and circle $\mathrm { C } _ { 3 }$ is obtained. By repeating this process infinitely many times, the sequence of circles $C _ { 1 } , C _ { 2 } , C _ { 3 } , \cdots$ is obtained. For each positive integer $\mathbf { k }$, let $\mathbf { A } _ { \mathbf { k } }$ be the area bounded by circle $\mathbf { C } _ { \mathbf { k } }$ in square units. Accordingly, what is the result of the sum $$\sum _ { k = 1 } ^ { \infty } A _ { k }$$ ? A) $\frac { 3 \pi } { 2 }$ B) $\frac { 4 \pi } { 3 }$ C) $\frac { 5 \pi } { 4 }$ D) $\frac { 6 \pi } { 5 }$ E) $\frac { 9 \pi } { 8 }$
An equilateral triangle is inscribed in circle $\mathrm { C } _ { 1 }$ with radius 1 unit as shown in the figure. Let $\mathrm { C } _ { 2 }$ be the circle passing through the midpoints of the sides of this triangle.\\
The same operation is performed for circle $\mathrm { C } _ { 2 }$ and circle $\mathrm { C } _ { 3 }$ is obtained. By repeating this process infinitely many times, the sequence of circles $C _ { 1 } , C _ { 2 } , C _ { 3 } , \cdots$ is obtained.
For each positive integer $\mathbf { k }$, let $\mathbf { A } _ { \mathbf { k } }$ be the area bounded by circle $\mathbf { C } _ { \mathbf { k } }$ in square units. Accordingly, what is the result of the sum
$$\sum _ { k = 1 } ^ { \infty } A _ { k }$$
?\\
A) $\frac { 3 \pi } { 2 }$\\
B) $\frac { 4 \pi } { 3 }$\\
C) $\frac { 5 \pi } { 4 }$\\
D) $\frac { 6 \pi } { 5 }$\\
E) $\frac { 9 \pi } { 8 }$