smc

Note the areas of these circles is $1\pi$, $\frac{\pi }{4}$, $\frac{\pi }{16},\dots$. The sum of these areas will thus be $\pi \left(1+\frac{1}{4}+\frac{1}{16}+\dots \right)$. We use the formula for an infinite geometric series to get the sum of the areas will be $\pi \left(\frac{1}{1-\frac{1}{4}}\right)$, or $\frac{4\pi }{3}$. $\boxed{\frac{4\pi }{3}}$