Answer
$2\pi R^2$
Work Step by Step
Step 1. Build up a sequence of disk areas: the area for the firs disk is $a_1=\pi R^2$
Step 2. The second layer contains two disks of radius $\frac{R}{2}$ each, and the area is $a_2=2\times\pi(\frac{R}{2})^2=\frac{1}{2}\pi R^2=\frac{1}{2}a_1$
Step 3. The third layer contains four disks of radius $\frac{R}{4}$ each, and the area is $a_3=4\times\pi (\frac{R}{4})^2=\frac{1}{4}\pi R^2=\frac{1}{2}a_2$
Step 4. Repeat the process and we can see that the sequence is of geometric with $r=\frac{1}{2}$
Step 5. We can find the total area for infinite number of circles as $A_n=\pi R^2\times\frac{1}{1-1/2}=2\pi R^2$
